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THEORETICAL  ELEMENTS 

OF 

ELECTRICAL  ENGINEERING 


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THEORETICAL  ELEMENTS 

OF 

ELECTRICAL  ENGINEERING 


BY 
CHARLES  PROTEUS  STEINMETZ,  A.M.,  PH.D. 


FOURTH  EDITION 
THOROUGHLY  REVISED  AND  ENTIRELY  RESET 


FOURTH  IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC 

239  WEST  39TH  STREET.    NEW  YORK 


LONDON:  HILL  PUBLISHING  CO.,  LTD. 
6  &  8  BOUVERIE  ST.,  E.  C. 

1915 


e- 


Engineering 
library 


COPYRIGHT,  1909,  1915,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY,  INC. 


THE    MAPI.E     PRTJSS     YORK    PA 


PREFACE  TO  FIRST  EDITION 

THE  first  part  of  the  following  volume  originated  from  a 
series  of  University  lectures  which  I  once  promised  to  deliver. 
This  part  can,  to  a  certain  extent,  be  considered  as  an  intro- 
duction to  my  work  on  "Theory  and  Calculation  of  Alternating 
Current  Phenomena,"  leading  up  very  gradually  from  the  ordi- 
nary sine  wave  representation  of  the  alternating  current  to  the 
graphical  representation  by  polar  coordinates,  from  there  to 
rectangular  components  of  polar  vectors,  and  ultimately  to  the 
symbolic  representation  by  the  complex  quantity.  The  present 
work  is,  however,  broader  in  its  scope,  in  so  far  as  it  comprises 
the  fundamental  principles  not  only  of  alternating,  but  also  of 
direct  currents. 

The  second  part  is  a  series  of  monographs  of  the  more  impor- 
tant electrical  apparatus,  alternating  as  well  as  direct  current. 
It  is,  in  a  certain  respect,  supplementary  to  "Alternating  Current 
Phenomena."  While  in  the  latter  work  I  have  presented  the 
general  principles  of  alternating  current  phenomena,  in  the  pres- 
ent volume  I  intended  to  give  a  specific  discussion  of  the  par- 
ticular features  of  individual  apparatus.  In  consequence  thereof, 
this  part  of  the  book  is  somewhat  less  theoretical,  and  more 
descriptive,  my  intention  being  to  present  the  most  important 
electrical  apparatus  in  all  their  characteristic  features  as  regard 
to  external  and  internal  structure,  action  under  normal  and  ab- 
normal conditions,  individually  and  in  connection  with  other  ap- 
paratus, etc. 

I  have  restricted  the  work  to  those  apparatus  which  experi- 
ence has  shown  as  of  practical  importance,  and  give  only  those 
theories  and  methods  which  an  extended  experience  in  the  de- 
sign and  operation  has  shown  as  of  practical  utility.  I  con- 
sider this  the  more  desirable  as,  especially  of  late  years,  electri- 
cal literature  has  been  haunted  by  so  many  theories'  (for  instance 
of  the  induction  machine)  which  are  incorrect,  or  too  compli- 
cated for  use,  or  valueless  in  practical  application.  In  the  class 
last  mentioned  are  most  of  the  graphical  methods,  which,  while 
they  may  give  an  approximate  insight  in  the  inter-relation  of 


520444 


vi     '  PREFACE 

phenomena,  fail  entirely  in  engineering  practice  owing  to  the 
great  difference  in  the  magnitudes  of  the  vectors  in  the  same 
diagram,  and  to  the  synthetic  method  of  graphical  representa- 
tion, which  generally  require  one  to  start  with  the  quantity 
which  the  diagram  is  intended  to  determine. 

I  originally  intended  to  add  a  chapter  on  Rectifying  Apparatus, 
as  arc  light  machines  and  alternating  current  rectifiers,  but  had 
to  postpone  this,  due  to  the  incomplete  state  of  the  theory  of 
these  apparatus. 

The  same  notation  has  been  used  as  in  the  Third  Edition  of 
"  Alternating  Current  Phenomena,"  that  is,  vector  quantities 
denoted  by  dotted  capitals.  The  same  classification  and  nomen- 
clature have  been  used  as  given  by  the  report  of  the  Standardiz- 
ing Committee  of  the  American  Institute  of  Electrical  Engineers. 

CHARLES  PROTEUS  STEINMETZ. 
SCHENECTADY,  N.  Y.,  May  1st,  1901. 


PREFACE  TO  THIRD  EDITION 

NEARLY  eight  years  have  elapsed  since  the  appearance  of  the 
second  edition,  during  which  time  the  book  has  been  reprinted 
without  change,  and  a  revision,  therefore,  became  greatly  desired. 

It  was  gratifying,  however,  to  find  that  none  of  the  contents 
of  the  former  edition  had  to  be  dropped  as  superseded  or  anti- 
quated. However,  very  much  new  material  had  to  be  added. 
During  these  eight  years  the  electrical  industry  has  progressed 
at  least  as  rapidly  as  in  any  previous  period,  and  apparatus  and 
phenomena  which  at  the  time  of  the  second  edition  were  of 
theoretical  interest  only,  or  of  no  interest  at  all,  have  now  as- 
sumed great  industrial  importance,  as  for  instance  the  single- 
phase  commutator  motor,  the  control  of  commutation  by  corn- 
mutating  poles,  etc. 

Besides  rewriting  and  enlarging  numerous  paragraphs  through- 
out the  text,  a  number  of  new  sections  and  chapters  have  been 
added,  on  alternating-current  railway  motors,  on  the  control  of 
commutation  by  commutating  poles  ("interpoles"),  on  con- 
verter heating  and  output,  on  converters  with  variable  ratio  of 
conversion  (" split-pole  converters"),  on  three- wire  generators 
and  converters,  short-circuit  currents  of  alternators,  stability 
and  regulation  of  induction  motors,  induction  generators,  etc. 

In  conformity  with  the  arrangement  used  in  my  other  books, 
the  paragraphs  of  the  text  have  been  numbered  for  easier  refer- 
ence and  convenience. 

When  reading  the  book,  or  using  it  as  text-book,  it  is  recom- 
mended : 

After  reading  the  first  or  general  part  of  the  present  volume, 
to  read  through  the  first  17  chapters  of  "Theory  and  Calculation 
of  Alternating  Current  Phenomena/'  omitting,  however,  the 
mathematical  investigations  as  far  as  not  absolutely  required 
for  the  understanding  of  the  text,  and  then  to  take  up  the  study 
of  the  second  part  of  the  present  volume,  which  deals  with 
special  apparatus.  When  reading  this  second  part,  it  is  recom- 
mended to  parallel  its  study  with  the  reading  of  the  chapter  of 
" Alternating  Current  Phenomena"  which  deals  with  the  same 

vii 


viii  PREFACE 

subject  in  a  different  manner.     In  this  way  a  clear  insight  into 
the  nature  and  behavior  of  apparatus  will  be  imparted. 

Where  time  is  limited,  a  large  part  of  the  mathematical  dis- 
cussion may  be  skipped  and  in  that  way  a  general  review  of  the 
material  gained. 

Great  thanks  are  due  to  the  technical  staff  of  the  McGraw- 
Hill  Book  Company,  which  has  spared  no  effort  to  produce  the 
third  edition  in  as  perfect  and  systematic  a  manner  as  possible, 
and  to  the  numerous  engineers  who  have  greatly  assisted  me  by 
pointing  out  typographical  and  other  errors  in  the  previous 
edition. 

CHARLES  PROTEUS  STEINMETZ. 
SCHENECTADY,  September,  1909. 


PREFACE  TO  THE  FOURTH  EDITION 

With  the  fourth  edition,  "  Theoretical  Elements  of  Electrical 
Engineering"  has  been  radically  revised  and  practically  rewritten. 
Since  1897  and  1898,  when  the  first  editions  of  "  Alternating 
Current  Phenomena"  and  "Theoretical  Elements"  appeared, 
electrical  engineering  has  enormously  expanded  and  diversified. 
New  material  thus  had  to  be  added  to  the  successive  editions 
until  now  it  has  become  utterly  impossible  to  deal  with  the  sub- 
ject matter  adequately  within  the  limited  scope  of  the  two 
books.  Therefore  in  the  present  edition  everything  beyond  the 
most  fundamental  principles  of  general  theory  and  special  ap- 
paratus has  been  withdrawn,  to  make  room  for  the  adequate  rep- 
resentation of  the  theoretical  elements  of  present-day  electrical 
engineering.  The  same  will  be  done  in  the  new  edition  of  "Alter- 
nating Current  Phenomena,"  which  is  in  preparation,  and  the 
material,  which  thus  does  not  find  room  any  more  in  these  two 
books,  together  with  such  additional  matters  as  the  development 
of  electrical  engineering  requires,  will  be  collected  in  a  third 
volume. 

In  the  present  edition,  the  crank  diagram  of  vector  represen- 
tation, and  the  symbolic  method  based  on  it,  which  denotes  the 
inductive  impedance  by  Z  —  r  +  jx,  has  been  adopted  in  con- 
formity with  the  decision  of  the  International  Electrical  Congress 
of  Turin.  This  crank  diagram  is  somewhat  inferior  in  utility 
to  the  polar  diagram  used  in  the  previous  editions,  since  it  is 
limited  to  sine  waves.  I  believe  it  was  adopted  without  sufficient 
consideration  of  the  relative  merits.  Nevertheless  the  advan- 
tage of  the  use  of  the  same  vector  representation  in  all  elementary 
text-books  on  electrical  engineering,  seems  to  me  to  outweigh  the 
advantage  of  the  polar  diagram  resulting  from  its  ability  to  rep- 
resent waves  which  are  not  sines,  while  in  advanced  electrical 
engineering  both  representations  will  have  to  remain  in  use. 

CHARLES  P.  STEINMETZ. 
SCHENECTADY,  N.  Y.,  October,  1915. 


IX 


CONTENTS 

PART  I 
GENERAL  THEORY 

PAGE 

1.  Magnetism  and  Electric  Current.  1 

2.  Magnetism  and  E.M.F.  9 

3.  Generation  of  E.M.F.  11 

4.  Power  and  Effective  Values.  15 

5.  Self-Inductance  and  Mutual  Inductance.  21 

6.  Self-Inductance  of  Continuous-Current  Circuits.  24 

7.  Inductance  in  Alternating-Current  Circuits.  31 

8.  Power  in  Alternating-Current  Circuits.  39 

9.  Vector  Diagrams.  41 

10.  Hysteresis  and  Effective  Resistance.  48 

11.  Capacity  and  Condensers.  54 

12.  Impedance  of  Transmission  Lines.  57 

13.  Alternating-Current  Transformer.  67 

14.  Rectangular  Coordinates.  77 

15.  Load  Characteristic  of  Transmission  Line.  85 

16.  Phase  Control  of  Transmission  Lines.  90 

17.  Impedance  and  Admittance.  98 

18.  Equivalent  Sine  Waves.  106 

19.  Fields  of  Force. 

20.  Nomenclature.  118 

PART  II 

SPECIAL  APPARATUS 

INTRODUCTION.  121 

A.  SYNCHRONOUS  MACHINES. 

I.  General.  126 

II.  Electromotive  Forces.  128 

III.  Armature  Reaction.  130 

IV.  Self-Inductance.  133 
V.  Synchronous  Reactance.  136 

VI.  Characteristic  Curves  of  Alternating-Current  Generator.  138 

VII.  Synchronous  Motor.  141 

VIII.  Characteristic  Curves  of  Synchronous  Motor.  143 

IX.  Magnetic  Characteristic  or  Saturation  Curve.  147 

X.  Efficiency  and  Losses.  149 

XI.  Unbalancing  of  Polyphase  Synchronous  Machines.  150 

XII.  Starting  of  Synchronous  Motors.  151 

Xin.  Parallel  Operation.  153 

XIV.  Division  of  Load  in  Parallel  Operation.  154 

xi 


xii  CONTENTS 

SYNCHRONOUS  MACHINES  (continued). 

PAGE 

XV.  Fluctuating  Cross-Currents  in  Parallel  Operation.  155 
XVI.  High     Frequency;     Cross-Currents     between     Synchronous 

Machines.  159 

XVII.  Short-Circjiit  Currents  of  Alternators.  160 

B.  DIRECT-CURRENT  COMMUTATING  MACHINES. 

I.  General.  166 

II.  Armature  Winding.  168 

III.  Generated  Electromotive  Forces.  177 

IV.  Distribution  of  Magnetic  Flux.  178 
V.  Effect  of  Saturation  on  Magnetic  Distribution.  182 

VI.  Effect  of  Commutating  Poles.  184 

VII.  Effect  of  Slots  on  Magnetic  Flux.  190 

VIII.  Armature  Reaction.  193 

IX.  Saturation  Curves.  194 

X.  Compounding.  196 

XI.  Characteristic  Curves.  198 

XII.  Efficiency  and  Losses.  198 

XIII.  Commutation.  199 

XIV.  Types  of  Commutating  Machines.  206 

A.  Generators.     Separately  excited  and  Magneto,  Shunt, 

Series,  Compound.  208 

B.  Motors.     Shunt,  Series,  Compound.  215 
XV.  Appendix.     Alternating-Current  Commutator  Motor.  218 

C.  SYNCHRONOUS  CONVERTERS. 

I.  General.  223 

II.  Ratio  of  E.M.Fs.  and  of  Currents.  224 

III.  Variation  of  the  Ratio  of  E.M.Fs.  231 

•  IV.  Armature  Current  and  Heating.  232 

V.  Armature  Reaction.  245 

VI.  Reactive  Currents  and  Compounding.  250 

VII.  Variable  Ratio  Converters  (Split-Pole  Converters).  252 

V11I.  Starting.  .                  253 

IX.  Inverted  Converters.  255 

X.  Frequency.  257 

XI.  Double-Current  Generators.  259 

XII.  Conclusion.  261 

XIII.  Appendix.     Direct-Current  Converter.  262 

XIV.  Three- Wire  Generator  and  Converter.  270 

D.  ALTERNATING-CURRENT  TRANSFORMER. 

I.  General.  277 

II.  Excitation.  279 


CONTENTS  xiii 
ALTERNATING-CURRENT  TRANSFORMER  (continued). 

PAGE 

III.  Losses  and  Efficiency.  280 

IV.  Regulation.  285 
V.  Short-Circuit  Current.  293 

VI.  Heating  and  Ventilation.  294 

VII.  Types  of  Transformers.  295 

VIII.  Auto-Transformers.  299 

IX.  Reactors.  302 

E.  INDUCTION  MACHINES. 

I.  General.  306 
II.  Polyphase  Induction  Motor. 

1.  Introduction.  310 

2.  Calculation.  311 

3.  Load  and  Speed  Curves.  317 

4.  Effect  of  Armature  Resistance  and  Starting.  322 

III.  Single-phase  Induction  Motor. 

1.  Introduction.  326 

2.  Load  and  Speed  Curves.  329 

3.  Starting  Devices  of  Single-phase  Motors.  333 

4.  Acceleration  with  Starting  Device.  338 

IV.  Induction  Generator. 

1.  Introduction.  340 

2.  Constant  Speed  Induction  or  Asynchronous  Generator.  342 

3.  Power  Factor  of  Induction  Generator.  343 
V.  Induction  Booster.  349 

VI.  Phase  Converter.  351 
VII.  Frequency  Converter  or  General  Alternating-Current  Trans- 
former. 354 
VIII.  Concatenation  of  Induction  Motors.  359 

INDEX  363 


PART  I 
GENERAL  THEORY 


1.  MAGNETISM  AND  ELECTRIC  CURRENT 

1.  A  magnet  pole  attracting  (or  repelling)  another  magnet 
pole  of  equal  strength  at  unit  distance  with  unit  force1  is  called 
a  unit  magnet  pole. 

The  space  surrounding  a  magnet  pole  is  called  a  magnetic  field 
of  force,  or  magnetic  field. 

The  magnetic  field  at  unit  distance  from  a  unit  magnet  pole 
is  called  a  unit  magnetic  field,  and  is  represented  by  one  line  of 
magnetic  force  (or  shortly  "one  line")  per  square  centimeter, 
and  from  a  unit  magnet  pole  thus  issue  a  total  of  4  TT  lines  of 
magnetic  force. 

The  total  number  of  lines  of  force  issuing  from  a  magnet  pole 
is  called  its  magnetic  flux. 

The  magnetic  flux  $  of  a  magnet  pole  of  strength  m  is, 

<£  =  4  irm. 

At  the  distance  I  from  a  magnet  pole  of  strength  m,  and 
therefore  of  flux  $>  =  4  xw,  assuming  a  uniform  distribution  in 
all  directions,  the  magnetic  field  has  the  intensity, 

n  - 


since  the  3>  lines  issuing  from  the  pole  distribute  over  the  area 
of  a  sphere  of  radius  I,  that  is,  the  area  4  irl2. 

A  magnetic  field  of  intensity  H  exerts  upon  a  magnet  pole 
of  strength  m  the  force, 

mH. 

Thus  two  magnet  poles  of  strengths  mi  and  mz,  and  distance 
I  from  each  other,  exert  upon  each  other  the  force, 


1  That  is,  at  1  cm.  distance  with  such  force  as  to  give  to  the  mass  of  1  gram 
the  acceleration  of  1  cm.  per  second. 

1 


2         *:LE' VENTS  OF  ELECTRICAL  ENGINEERING 

2.  Electric   currents   produce   magnetic   fields   also;   that   is, 
the  space  surrounding  the  conductor  carrying  an  electric  current 
is  a  magnetic  field,  which  appears  and  disappears  and  varies 
with  the  current  producing  it,  and  is  indeed  an  essential  part 
of  the  phenomenon  called  an  electric  current. 

Thus  an  electric  current  represents  a'  magnetomotive  force 
(m.m.f.). 

The  magnetic  field  of  a  straight  conductor,  whose  return 
conductor  is  so  far  distant  as  not  to  affect  the  field,  consists  of 
lines  of  force  surrounding  the  conductor  in  concentric  circles. 
The  intensity  of  this  magnetic  field  is  directly  proportional 
to  the  current  strength  and  inversely  proportional  to  the  dis- 
tance from  the  conductor. 

Since  the  lines  of  force  of  the  magnetic  field  produced  by 
an  electric  current  return  into  themselves,  the  magnetic  field 
is  a  magnetic  circuit.  Since  an  electric  current,  at  least  a  steady 
current,  can  exist  only  in  a  closed  circuit,  electricity  flows 
in  an  electric  circuit.  The  magnetic  circuit  produced  by  an 
electric  current  surrounds  the  electric  circuit  through  which 
the  electricity  flows,  and  inversely.  That  is,  the  electric  circuit 
and  the  magnetic  circuit  are  interlinked  with  each  other. 

Unit  current  in  an  electric  circuit  is  the  current  which  produces 
in  a  magnetic  circuit  of  unit  length  the  field  intensity  4?r,  that 
is,  produces  as  many  lines  of  force  per  square  centimeter  as 
issue  from  a  unit  magnet  pole. 

In  unit  distance  from  an  electric  conductor  carrying  unit 
current,  that  is,  in  a  magnetic  circuit  of  length  2?r,  the  field 

4-7T 

intensity  is  ~ —  —  2,  and  in  the  distance  2  the  field  intensity 

is  unity;  that  is,  unit  current  is  the  current  which,  in  a  straight 
conductor,  whose  return  conductor  is  so  far  distant  as  not  to 
affect  its  magnetic  field,  produces  field  intensity  2  in  unit  distance 
from  the  conductor. 

One-tenth  of  unit  current  is  the  practical  unit,  called  one  ampere. 

3.  One  ampere  in  an  electric  circuit  or  turn,  that  is,   one 
ampere-turn,  thus  produces  in  a  magnetic  circuit  of  unit  length 
the  field  intensity  0.4  w,  and  in  a   magnetic  circuit    of  length 

0.4  TT 
I  the  field  intensity  — '-j — ,  and  F  ampere-turns  produce  in  a 

magnetic  circuit  of  length  I  the  field  intensity: 

„       0.4  irFv          tf 

H  =    — j —  lines  of  force  per  sq.  cm. 


MAGNETISM  AND  ELECTRIC  CURRENT  3 

regardless  whether  the  F  ampere-turns  are  due  to  F  amperes 

F 

in  a  single  turn,  or  1  amp.  in  F  turns,  or  —  amperes  in  n  turns. 

F,  that  is,  the  product  of  amperes  and  turns,  is  called  magneto- 
motive force  (m.m.f.). 

The  m.m.f.  per  unit  length  of  magnetic  circuit,  or  ratio, 

_  _  m.m.f.  _ 
'  "  "  length  of  magnetic  circuit  » 
is  called  the  magnetizing  force,  or  magnetic  gradient. 

Hence,  m.m.f.  is  expressed  in  ampere-turns;  magnetizing 
force  in  ampere-turns  per  centimeter  (or  in  practice  frequently 
ampere-turns  per  inch),  field  intensity  in  lines  of  magnetic  force 
per  square  centimeter. 

At  the  distance  I  from  the  conductor  of  a  loop  or  circuit  of 
F  ampere-turns,  whose  return  conductor  is  so  far  distant  as  not 
to  affect  the  field,  assuming  the  m.m.f.  =  F,  since  the  length  of 
the  magnetic  circuit  =  2  irl,  we  obtain  as  the  magnetizing  force, 

:    '-« 

and  as  the  field  intensity, 

0  2  F 
H  =  0.4  TT    =  2^- 


4.  The  magnetic  field  of  an  electric  circuit  consisting  of  two 
parallel  conductors  (or  any  number  of  conductors,  in  a  poly- 
phase system),  as  the  two  wires  of  a  transmission  line,  can  be 
considered  as  the  superposition  of  the  separate  fields  of  the 
conductors  (consisting  of  concentric  circles).  Thus,  if  there  are 
I  amperes  in  a  circuit  consisting  of  two  parallel  conductors 
(conductor  and  return  conductor),  at  the  distance  li  from  the 
first  and  h  from  the  second  conductor,  the  respective  field 
intensities  are, 


- 

l  ~  ~TT 

and 

"-T 

and  the  resultant  field  intensity,  if  r  =   angle  between  the  direc- 
tions of  the  two  fields, 


H  =  \/#  i2  +  #22  +  2  #!#2  COST, 


COST. 


4         ELEMENTS  OF  ELECTRICAL  ENGINEERING 

In  the  plane  of  the  conductors,  where  the  two  fields  are  in 
the  same  or  opposite  direction,  the  resultant  field  intensity  is, 


7     7 

M2 

where  the  plus  sign  applies  to  the  space  between,  the  minus 
sign  the  space  outside  of  the  conductors. 

The  resultant  field  of  a  circuit  of  parallel  conductors  con- 
sists of  excentric  circles,  interlinked  with  the  conductors,  and 
crowded  together  in  the  space  between  the  conductors  as  shown 
in  Fig.  1  by  drawn  lines. 


FIG.  1. — Magnetic  field  of  parallel  conductors. 

The  magnetic  field  in  the  interior  of  a  spiral  (solenoid,  helix, 
coil)  carrying  an  electric  current  consists  of  straight  lines. 

5.  If  a  conductor  is  coiled  in  a  spiral  of  I  centimeter  axial 

N 
length  of  spiral,  and  N  turns,  thus  n  =  j-  turns  per  centimeter 

length  of  spiral,  and  /  =  current,  in  amperes,  in  the  conductor, 
the  m.m.f.  of  the  spiral  is 

F  =  IN, 

and  the  magnetizing  force  in  the  middle  of  the  spiral,  assuming, 
the  latter  of  very  great  length, 

T  N   T 

f  =  nI  =  TI, 

thus  the  field  intensity  in  the  middle  of  the  spiral  or  solenoid, 

H  =  0.4  TT/ 
=  0.4 


MAGNETISM  AND  ELECTRIC  CURRENT  5 

Strictly  this  is  true  only  in  the  middle  part  of  a  spiral  of 
such  length  that  the  m.m.f.  consumed  by  the  external  or  mag- 
netic return  circuit  of  the  spiral  is  negligible  compared  with 
the  m.m.f.  consumed  by  the  magnetic  circuit  in  the  interior 
of  the  spiral,  or  in  an  endless  spiral,  that  is,  a  spiral  whose  axis 
curves  back  into  itself,  as  a  spiral  whose  axis  is  curved  in  a 
circle. 

Magnetomotive  force  F  applies  to  the  total  magnetic  circuit, 
or  part  of  the  magnetic  circuit.  It  is  measured  in  ampere- 
turns. 

Magnetizing  force  /  is  the  m.m.f.  per  unit  length  of  mag- 
netic circuit.  It  is  measured  in  ampere-turns  per  centimeter. 

Field  intensity  H  is  the  number  of  lines  of  force  per  square 
centimeter. 

If  I  =  length  of  the  magnetic  circuit  or  a  part  of  the  magnetic 
circuit, 

F  =  V,  f  =  j, 

H  =  0.4  »/  /  =     H 


0.47T' 

=  1.257/  /=  0.796  #. 

6.  The  preceding  applies  only  to  magnetic  fields  in  air  or 
other  unmagnetic  materials. 

If  the  medium  in  which  the  magnetic  field  is  established  is  a 
"magnetic  material,"  the  number  of  lines  of  force  per  square 
centimeter  is  different  and  usually  many  times  greater.  (Slightly 
less  in  diamagnetic  materials.) 

The  ratio  of  the  number  of  lines  of  force  in  a  medium,  to  the 
number  of  lines  of  force  which  the  same  magnetizing  force  would 
produce  in  air  (or  rather  in  a  vacuum),  is  called  the  permeability 
or  magnetic  conductivity  /*  of  the  medium. 

The  number  of  lines  of  force  per  square  centimeter  in  a  mag- 
netic medium  is  called  the  magnetic  induction  B.  The  number 
of  lines  of  force  produced  by  the  same  magnetizing  force  in  air, 
or  rather,  in  the  vacuum,  is  called  the  field  intensity  H. 

In  air,  magnetic  induction  B  and  field  intensity  H  are  equal. 

As  a  r-ule,  the  magnetizing  force  in  a  magnetic  circuit  is 
changed  by  the  introduction  of  the  magnetic  material,  due  to  the 
change  of  distribution  of  the  magnetic  flux. 

The  permeability  of  air  =  1  and  is  constant. 


6         ELEMENTS  OF  ELECTRICAL  ENGINEERING . 

The  permeability  of  iron  and  other  magnetic  materials  varies 
with  the  magnetizing  force  between  a  little  above  1  and  values 
beyond  10,000  in  soft  iron. 

The  magnetizing  force  /  in  a  medium  of  permeability  /*  pro- 
duces the  field  intensity  H  =  0.4  irf  and  the  magnetic  induction 
B  =  0.4  TTflf. 

EXAMPLES 

7.  (1)  A  pull  of  2  grams  at  4  cm.  radius  is  required  to  hold  a 
horizontal  bar  magnet  12  cm.  in  length,  pivoted  at  its  center, 
in  a  position  at  right  angles  to  the  magnetic  meridian.     What 
is  the  intensity  of  the  poles  of  the  magnet,  and  the  number  of 
lines  of  magnetic  force  issuing  from  each  pole,  if  the  horizontal 
intensity  of  the  terrestrial   magnetic   field   H  =  0.2,    and   the 
acceleration  of  gravity  =  980? 

The  distance  between  the  poles  of  the  bar  magnet  may  be 
assumed  as  five-sixths  of  its  length. 

Let  m  =  intensity  of  magnet  poles.  I  =  5  is  the  radius  on 
which  the  terrestrial  magnetism  acts. 

Thus  2mHl  =  2'm  =  torque  exerted  by  the  terrestrial 
magnetism. 

2  grams  weight  =  2  X  980  =  1960  units  of  force.  These  at 
4  cm.  radius  give  the  torque  4  X  I960  =  7840  g  cm. 

Hence  2m  =  7840. 

m  =  3920  is  the  strength  of  each  magnet  pole  and 

3>  =  4  Trm  =  49,000,  the  number  of  lines  of  force  issuing  from 
each  pole. 

8.  (2)    A   conductor   carrying   100   amp.  runs  in  the   direc- 
tion of  the  magnetic  meridian.     What  position  will  a   compass 
needle  assume,  when  held  below  the  conductor  at  a  distance  of 
50  cm.,  if  the  intensity  of  the  terrestrial  magnetic  field  is  0.2? 

The  intensity   of  the   magnetic   field   of   100   amp.,  50  cm. 

027  100 

from  the  conductor,  is  H  =  — ^ —  =  0.2  X  -^r  =  0.4,  the  direc- 
tion is  at  right  angles  to  the  conductor,  that  is,  at  right  angles 
to  the  terrestrial  magnetic  field. 

If  T  =  angle  between  compass  needle  and  the  north  pole  of 
the  magnetic  meridian,  10  =  length  of  needle,  m  =  intensity  of 
its  magnet  pole,  the  torque  of  the  terrestrial  magnetism  is  Hmlo 
sin  T  =  0.2  mlo  sin  r,  the  torque  of  the  current  is 

0.2/raZocosr 
cos  r  =  -       — —      -  =  0.4 


MAGNETISM  AND  ELECTRIC  CURRENT  7 

In  equilibrium,  0.2  mlQ  sin  r  =  0.4  mlQ  cos  r,  or  tan  r  =  2, 
r  =  63.4°. 

9.  (3)  What  is  the- total  magnetic  flux  per  I  =  1000  m.  length, 
passing  between  the  conductors  of  a  long  distance  transmission 
line  carrying  7  amperes  of  current,  if  Id  =  0.82  cm.  is  the  diam- 
eter of  the  conductors  (No.  0  B.  &  S.),  18  =  45  cm.  the  spacing 
or  distance  between  them? 


FIG.  2. — Diagram  of  transmission  line  for  inductance  calculation. 

At  distance  lr  from  the  center  of  one  of  the  conductors  (Fig.  2), 
the  length  of  the  magnetic  circuit  surrounding  this  conductor 
is  2irlr)  the  m.m.f.,  7  ampere-turns;  thus  the  magnetizing  force 

/  =  s— r>    and  the  field  intensity  H  =  0.4  irf  =  -^ — >     and  the 

A  TTlr  Lr 

flux  in  the  zone  dlr  is  d$  =  -  — j ->    and  the  total  flux  from  the 

surface  of  the  conductor  to  the  next  conductor  is, 

).2  Ildlr 


0.27/Floge  Zrj£  =  0.2/nog.^- 


The  same  flux  is  produced  by  the  return  conductor  in  the 
same  direction,  thus  the  total  flux  passing  between  the  trans- 
mission wires  is, 

2  $  =  0.4  II  log,  p 

I'd 

or  per  1000  m.  =  105  cm.  length, 

QO 
2  $  =  0.4  X  105  /  log,  TT^  =  0.4  X  105  X  4.70  I  =  0.188  X  106  7, 


8         ELEMENTS  OF  ELECTRICAL  ENGINEERING 

or  0.188  /  megalines  or  millions  of  lines  per  line  of  1000  m. 
of  which  0.094  /  megalines  surround  each  of  the  two  conductors. 
10.  (4)  In  an  alternator  each  pole  has  to  carry  6.4  millions 
of  lines,  or  6.4  megalines  magnetic  flux.  How  many  ampere- 
turns  per  pole  are  required  to  produce  this  flux,  if  the  magnetic 


2  8        10         2       14         6 


FIG.  3. — Magnetization  curves  of  various  irons. 

circuit  in  the  armature  of  laminated  iron  has  the  cross  section 
of  930  sq.  cm.  and  the  length  of  15  cm.,  the  air-gap  between 
stationary  field  poles  and  revolving  armature  is  0.95  cm.  in 
length  and  1200  sq.  cm.  in  section,  the  field  pole  is  26.3  cm. 
in  length  and  1075  sq.  cm.  in  section,  and  is  of  laminated  iron, 


MAGNETISM  AND  E.M.F.  9 

and  the  outside  return  circuit  or  yoke  has  a  length  per  pole  of 
20  cm.  and  2250  sq.  cm.  section,  and  is  of  cast  iron? 

The  magnetic  densities  are:  BI  =  6880  in  the  armature,  B2  = 
5340  in  the  air-gap,  B3  =  5950  in  the  field  pole,  and  B4  =  2850 
in  the  yoke.  The  permeability  of  sheet  iron  is  m  =  2550  at 
5i  =  6880,  MS  =  2380  at  B3  =  5950.  The  permeability  of  cast 

iron  is  /z4  =  280  at  B4  =  2850.  Thus  the  field  intensity/  H  =  -  j 
is:  Hi  =  2.7,  #2  =  5340,  H3  =  2.5,  H,  =  10.2. 

The  magnetizing  force  (/  =  {TT— )  is>  fl  =  2'15>  ^2  =  4250' 

/3  =  1.99,  /4  =  8.13  ampere-turns  per  centimeter.  Thus  the 
m.m.f.  (F  =  fl)  is:  Fi  =  32,  F2  =  4040,  F3  =  52,  F4  =  163,  or 
the  total  m.m.f.  per  pole  is 

F  =  Fi  +  F2  +  F3  +  ^4  =  4290  ampere-turns. 

The  permeability  p  of  magnetic  materials  varies  with  the 
density  B,  thus  tables  or  curves  have  to  be  used  for  these  quan- 
tities. Such  curves  are  usually  made  out  for  density  B  and 
magnetizing  force  /,  so  that  the  magnetizing  force  /  correspond- 
ing to  the  density  B  can  be  derived  directly  from  the  curve. 
Such  a  set  of  curves  is  given  in  Fig.  3. 

2.  MAGNETISM  AND  E.M.F. 

11.  In  an  electric  conductor  moving  relatively  to  a  magnetic 
field,  an  e.m.f.  is  generated  proportional  to  the  rate  of  cutting 
of  the  lines  of  magnetic  force  by  the  conductor. 

Unit  e.m.f.  is  the  e.m.f.  generated  in  a  conductor  cutting  one 
line  of  magnetic  force  per  second. 

108  times  unit  e.m.f.  is  the  practical  unit,  called  the  volt. 
Coiling  the  conductor  n  fold  increases  the  e.m.f.  n  fold,  by 

cutting  each  line  of  magnetic  force  n  times. 

In  a  closed  electric  circuit  the  e.m.f.  produces  an  electric 
current. 

The  ratio  of  e.m.f.  to  electric  current  produced  thereby  is 
called  the  resistance  of  the  electric  circuit. 

Unit  resistance  is  the  resistance  of  a  circuit  in  which  unit 
e.m.f.  produces  unit  current. 

109  times   unit  resistance  is  the  practical  unit,   called   the 
ohm. 


10       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

The  ohm  is  the  resistance  of  a  circuit,  in  which  1  volt 
produces  1  amp. 

The  resistance  per  unit  length  and  unit  section  of  a  conductor 
is  called  its  resistivity,  p. 

The  resistivity  p  is  a  constant  of  the  material,  varying  with 
the  temperature. 

The  resistance  r  of  a  conductor  of  length  I,  area  or  section  A, 

...  lp 

and  resistivity  p  is  r  =  -7" 

12.  If  the  current  in  the  electric  circuit  changes,  starts,  or 
stops,  the  corresponding  change  of  the  magnetic  field  of  the 
current  generates  an  e.m.f  in  the  conductor  carrying  the  current, 
which  is  called  the  e.m.f.  of  self-induction. 

If  the  e.m.f.  in  an  electric  circuit  moving  relatively  to  a 
magnetic  field  produces  a  current  in  the  circuit,  the  magnetic 
field  produced  by  this  current  is  called  its  magnetic  reaction. 

The  fundamental  law  of  self-induction  and  magnetic  reaction 
is  that  these  effects  take  place  in  such  a  direction  as  to  oppose 
their  cause  (Lentz's  law). 

Thus  the  e.m.f.  of  self-induction  during  an  increase  of  current 
is  in  the  opposite  direction,  during  a  decrease  of  current  in  the 
same  direction  as  the  e.m.f.  producing  the  current. 

The  magnetic  reaction  of  the  current  produced  in  a  circuit 
moving  out  of  a  magnetic  field  is  in  the  same  direction,  in  a 
circuit  moving  into  a  magnetic  field  in  opposite  direction  to  the 
magnetic  field. 

Essentially,  this  law  is  nothing  but  a  conclusion  from  the  law 
of  conservation  of  energy. 

EXAMPLES 

13.  (1)    An   electromagnet  is  placed  so  that  one  pole  sur- 
rounds the  other  pole  cylindrically  as  shown  in  section  in  Fig.  4, 
and  a  copper  cylinder  revolves  between  these  poles  at  3000  rev. 
per  min.     What  is  the  e.m.f.  generated  between  the  ends  of  this 
cylinder,  if  the  magnetic  flux  of  the  electromagnet  is   <£  =  25 
megalines? 

During  each  revolution  the  copper  cylinder  cuts  25  mega- 
lines.  It  makes  50  rev.  per  sec.  Thus  it  cuts  50  X  25  X  106  = 
12.5  X  108  lines  of  magnetic  flux  per  second.  Hence  the  gener- 
ated e.m.f.  is  E  =  12.5  volts. 


GENERATION  OF  E.M.F. 


11 


Such  a  machine  is  called  a  "  unipolar,"  or  more  properly  a 
" non-polar"  or  an  "acyclic,"  generator. 

14.  (2)  The  field  spools  of  the  20-pole  alternator  in  Section 
1,  Example  4,  are  wound  each  with  616  turns  of  wire  No.  7 
(B.  &  S.),  0.106  sq.  cm.  in  cross  section  and  160  cm.  mean  length 
of  turn.  The  20  spools  are  connected  in  series.  How  many 
amperes  and  how  many  volts  are  required  for  the  excitation  of 
this  alternator  field,  if  the  resistivity  of  copper  is  1.8  X  10~6 
ohms  per  cm.3 1 


FIG.  4. — Unipolar  generator. 

Since  616  turns  on  each  field  spool  are  used,  and  4280  ampere- 

4280 
turns  required,  the  current  is  fi1fi   =  6.95  amp. 

The  resistance  of  20  spools  of  616  turns  of  160  cm.  length, 
0.106  sq.  cm.  section,  and  1.8  X  10~6  resistivity  is, 


20  X  616  X  160  X  1.8  X  10~6 


=  33.2  ohms, 


0.106 
and  the  e.m.f.  required,  6.95  X  33.2  =  230  volts. 


i      3.  GENERATION  OF  E.M.F. 

15.  A  closed  conductor,  convolution  or  turn,  revolving  in  a 
magnetic  field,  passes  during  each  revolution  through  two 
positions  of  maximum  inclosure  of  lines  of  magnetic  force 
A  in  Fig.  5,  and  two  positions  of  zero  inclosure  of  lines  of  mag- 
netic force  B  in  Fig.  5. 

1  cm.3  refers  to  a  cube  whose  side  is  1  cm.,  and  should  not  be  confused 
with  cu.  cm. 


12       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Thus  it  cuts  during  each  revolution  four  times  the  lines  of 
force  inclosed  in  the  position  of  maximum  inclosure. 

If  3>  =  the  maximum  number  of  lines  of  force  inclosed  by 
the  conductor,  /  =  the  frequency  in  revolutions  per  second  or 
cycles,  and  n  =  number  of  convolutions  or  turns  of  the  con- 
ductor, the  lines  of  force  cut  per  second  by  the  conductor,  and 
thus  the  average  generated  e.m.f.  is, 

E  =  4  fn$  absolute  units, 
=  4fn3>  ID"8  volts. 


FIG.  5.  —  Generation  of  e.m.f. 

If  /  is  given  in  hundreds  of  cycles,  <£  in  megalines, 
E  =  4n$  volts. 


If  a  coil  revolves  with  uniform  velocity  through  a  uniform 
magnetic  field,  the  magnetism  inclosed  by  the  coil  at  any  instant 
is, 

$  COS  T 

where  $  =  the  maximum  magnetism  inclosed  by  the  coil  arid 
T  =  angle  between  coil  and  its  position  of  maximum  inclosure 
of  magnetism. 

The  e.m.f.  generated  in  the  coil,  which  varies  with  the  rate  of 
cutting  or  change  of  $  cos  T,  is  thus, 

e  =  EQ  sin  T, 

where  EQ  is  the  maximum  value  of  e.m.f.,  which  takes  place  for 
T  =  90°,  or  at  the  position  of  zero  inclosure  of  magnetic  flux 
since  in  this  position  the  rate  of  cutting  is  greatest. 

2 
Since  avg.  (sin  T)  =  -,  the  average  generated  e.m.f.  is, 


GENERATION  OF  E.M.F. 


13 


Since,  however,  we  found  above  that 

E  =  4  fn&  is  the  average  generated  e.m.f., 
it  follows  that 

EQ  =  2  irfn&  is  the  maximurft,  and 
e  =  2  7r/n$  sin  r  the  instantaneous  generated  e.m.f. 


The  interval  between  like  poles 
forms  360  electrical-space  de- 
grees, and  in  the  two-pole  model 
these  are  identical  with  the 
mechanical-space  degrees.  With 
uniform  rotation,  Fig.  6,  the 
space  angle,  r,  is  proportional  to 
time.  Time  angles  are  designated 
by  6,  and  with  uniform  rotation 
&  =  r,  T  being  measured  in  elec- 
trical-space  degrees. 


FIG.  6.  —  Generation  of  e.m.f. 
by  rotation. 


The  period  of  a  complete  cycle  is  360  time  degrees,  or  2  TT  or 


--  seconds.     In  the  two-pole  model  the  period  of  a  cycle  is  that  of 

one  complete  revolution,  and  in  a  2  np-pole  machine,  —  of  that 

np 

of  one  revolution. 

Thus,  6  =  2  wft 

e  =  2>jrfn3>  sin  2  irft. 

If  the  time  is  not  counted  from  the  moment  of  maximum 
Lnclosure  of  magnetic  flux,  but  ti  =  the  time  at  this  moment, 
we  have 

e  =  2  irfn$  sin  2  IT}  (t  -  ti) 
or,  e  =  2-jrfn&  sin  (6  —  0i), 

where  0i  =  2  irfli  is  the  angle  at  which  the  position  of  maxi- 
mum inclosure  of  magnetic  flux  takes  place,  and  is  called  its 
phase. 

These  e.m.fs.  are  alternating. 

If  at  the  moment  of  reversal  of  the  e.m.f.  the  connections 
between  the  coil  and  the  external  circuit  are  reversed,  the  e.m.f. 
in  the  external  circuit  is  pulsating  between  zero  and  EQ,  but 
has  the  same  average  value  E. 

If  a  number  of  coils  connected  in  series  follow  each  other 


H       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

successively  in  their  rotation  through  the  magnetic  field,  as  the 
armature  coils  of  a  direct-current  machine,  and  the  connections 
of  each  coil  with  the  external  circuit  are  reversed  at  the  moment 
of  reversal  of  its  e.m.f.,  their  pulsating  e.m.fs.  superimposed  in 
the  external  circuit  make  a  more  or  less  steady  or  continuous 
external  e.m.f. 

The  average  value  of  this  e.m.f.  is  the  sum  of  the  average  values 
of  the  e.m.fs.  of  the  individual  coils. 

Thus  in  a  direct-current  machine,  if  $  =  maximum  flux  in- 
closed per  turn,  n  =  total  number  of  turns  in  series  from  com- 
mutator brush  to  brush,  and  /  =  frequency  of  rotation  through 
the  magnetic  field. 

E  =  4/n$>  =  generated  e.m.f.    ($  in  megalines,  /  in 
hundreds  of  cycles  per  second). 

This  is  the  formula  of  the  direct-current  generator. 

EXAMPLES 

17.  (1)   A  circular  wire  coil  of  200  turns  and  40  cm.  mean 
diameter   is   revolved    around    a   vertical    axis.     What   is   the 
horizontal  intensity  of  the  magnetic  field  of  the  earth,  if  at  a 
speed  of  900  rev.  per  min.  the  average  e.m.f  generated  in  the  coil 
is  0.028  volt? 

4027T 

The  mean  area  of  the  coil  is  — j—  =  1255  sq.  cm.,  thus  the 

terrestrial  flux  inclosed  is  1255  H,  and  at  900  rev.  per  min.  or 
15  rev.  per  sec.,  this  flux  is  cut  4  X  15  =  60  times  per  second  by 
each  turn,  or  200  X  60  =  12,000  times  by  the  coil.  Thus  the 
total  number  of  lines  of  magnetic  force  cut  by  the  conductor 
per  second  is  12,000  X  1255  H  =  0.151  X  108  H,  and  the  average 
generated  e.m.f.  is  0.151  H  volts.  Since  this  is  =  0.028  volt, 
H  =  0.186. 

18.  (2)  In  a  550-volt  direct-current  machine  of  8  poles  and 
drum  armature,  running  at  500  rev.  per  min.,  the  average  vol- 
tage per  commutator  segment  shall  not  exceed  11,  each  armature 
coil  shall  contain  one  turn  only,  and  the  number  of  commutator 
segments  per  pole  shall  be  divisible  by  3,  so  as  to  use  the  machine 
as  three-phase  converter.     What  is  the  magnetic  flux  per  field 
pole? 

550  volts  at  11  volts  per  commutator  segment  gives  50,  or  as 
next  integer  divisible  by  3,  n  =  51  segments  or  turns  per  pole. 


POWER  AND  EFFECTIVE  VALUES  15 

8  poles  give  4  cycles  per  revolution,  500  rev.  per  min.  gives 
50%Q  =  8.33  rev.  per  sec.  Thus  the  frequency  is/  =  4  X  8.33 
=  33.3  cycles  per  second. 

The  generated  e.m.f.  is  E  =  550  volts,  thus  by  the  formula  of 
direct-current  generator, 

E  =  4/n<l>, 

or,  550  =  4  X  0.333  X  51  <l>, 

<i>  =  8.1  megalines  per  pole. 

19.  (3)  What   is  the  e.m.f.  generated  in  a  single  turn  of  a 
20-pole   alternator  running  at  200   rev.   per  min.,   through   a 
magnetic  field  of  6.4  megalines  per  pole? 

20  y  200 

The  frequency  is  /  =  V£~£  ~  =  33-3  cycles- 
^  X.  OU 

e  =  EQ  sin  r, 
EQ  =  2  irfn3>, 
$  =  6.4, 
n  =  l, 
f  =  0.333. 

Thus,         EQ  =  2  TT  X  0.333  X  6.4  =  13.4   volts  maximum,  or 
e  =  13.4  sin  0. 

4.  POWER  AND  EFFECTIVE  VALUES 

20.  The    power   of  the  continuous  e.m.f.  E  producing  con- 
tinuous current  /  is  P  =  El. 

The  e.m.f.  consumed  by  resistance  r  is  EI  =  7r,  thus  the 
power  consumed  by  resistance  r  is  P  =  72r. 

Either  EI  =  E,  then,  the  total  power  in  the  circuit  is  con- 
sumed by  the  resistance,  or  EI  <  E}  then  only  a  part  of  the 
power  is  consumed  by  the  resistance,  the  remainder  by  some 
counter  e.m.f.,  E  —  EI. 

If  an  alternating  current  i  =  I0  sin  6  passes  through  a  resist- 
ance r,  the  power  consumed  by  the  resistance  is, 

i*r  =  702r  sin2  0  =  ^r  C1  ~  cos  2  0), 
& 

thus  varies  with  twice  the  frequency  of  the  current,  between 
zero  and  70V. 

The  average  power  consumed  by  resistance  r  is, 

avg. 
since  avg.  (cos)  =  0. 


16       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Thus  the  alternating  current  i  =  IQ  since  0  consumes  in  a  resist- 
ance r  the  same  effect  as  a  continuous  current  of  intensity 


The  value  /  =  —7=  is  called  the  effective  value  of  the  alter- 

V2 

nating  current  i  =  IQ  sin  0;  since  it  gives  the  same  effect. 

ET 

Analogously  E  =  —i  is  the  effective  value  of  the  alternating 

V2 

e.m.f.,  e  =  EQ  sin  6. 

Since  E0  =  2  irfn$,  it  follows  that 


J' 

; 


=  4.44  fn&    ; 

is  the  effective  alternating  e.m.f.  generated  in  a  coil  of  turns  n 
rotating  at  a  frequency  of  /  (in  hundreds  of  cycles  per  second) 
through  a  magnetic  field  of  <E>  megalines  of  force. 

This  is  the  formula  of  the  alternating-current  generator. 

21.  The  formula  of  the  direct-current  generator, 

E  = 


holds  even  if  the  e.m.fs.  generated  in  the  individual  turns  are 
not  sine  waves,  since  it  is  the  average  generated  e.m.f. 
The  formula  of  the  alternating-current  generator, 

E  =  V2  *fn$, 

does  not  hold  if  the  waves  are  not  sine  waves,  since  the  ratios  of 
average  to  maximum  and  of  maximum  to  effective  e.m.f.  are 
changed. 

If  the  variation  of  magnetic  flux  is  not  sinusoidal,  the  effective 
generated  alternating  e.m.f.  is, 

E  =  7  \/2 


7  is  called  the  form  factor  of  the  wave,  and  depends  upon 
its  shape,  that  is,  the  distribution  of  the  magnetic  flux  in  the 
magnetic  field. 

Frequently  form  factor  is  defined  as  the  ratio  of  the  effect- 
ive to  the  average  value.  This  definition  is  undesirable  since  it 
gives  for  the  sine  wave,  which  is  always  considered  the  standard 
wave,  a  value  differing  from  one. 


POWER  AND  EFFECTIVE  VALUES  17 

EXAMPLES 

22.  (1)  In  a  star-connected  20-pole  three-phase  machine,  re- 
volving at  33.3  cycles  or  200  rev.  per  min.,  the  magnetic  flux 
per  pole  is  6.4  megalines.  The  armature  contains  one  slot  per 
pole  and  phase,  and  each  slot  contains  36  conductors.  All  these 
conductors  are  connected  in  series.  What  is  the  effective  e.m.f. 
per  circuit,  and  what  the  effective  e.m.f.  between  the  terminals 
of  the  machine? 

Twenty  slots  of  36  conductors  give  720  conductors,  or  360 
turns  in  series.  Thus  the  effective  e.m.f.  is,. 


=  4.44  X  0.333  X  360  X  6.4 
=  3400  volts  per  circuit. 


The  e.m.f.  between  the  terminals  of  a  star-connected  three- 
phase  machine  is  the  resultant  of  the  e.m.fs.  of  the  two  phases, 
which  differ  by  60  degrees,  and  is  thus  2  sin  60°  =  -\/3  times 
that  of  one  phase,  thus, 


E  = 

=  5900  volts  effective. 

23.  (2)  The  conductor  of  the  machine  has  a  section  of  0.22 
sq.  cm.  and  a  mean  length  of  240  cm.  per  turn.  At  a  resistivity 
(resistance  per  unit  section  and  unit  length)  of  copper  of  p  = 
1.8  X  10~6,  what  is  the  e.m.f.  consumed  in  the  machine  by  the 
resistance,  and  what  the  power  consumed  at  450  kw.  output? 

450  kw.  output  is  150,000  watts  per  phase  or  circuit,  thus 

150  000 
the  current  /  =     omn     =  44.2  amperes  effective. 


The  resistance  of  360  turns  of  240  cm.  length,  0.22  sq.  cm. 
section  and  1.8  X  10~6  resistivity,  is 

360  X  240  X  1.8  X  10~6 
r  =  -  —  -rT^  —  "  =  0.71  ohms  per  circuit. 

44.2  amp.  X  0.71  ohms  gives  31.5  volts  per  circuit  and  (44.2)2 
X  0.71  =  1400  watts  per  circuit,  or  a  total  of  3  X  1400  =  4200 
watts  loss. 

24.  (3)  What  is  the  self-inductance  per  wire  of  a  three- 
phase  line  of  14  miles  length  consisting  of  three  wires  No.  0 
(Id  =  0.82  cm.),  45  cm.  apart,  transmitting  the  output  of  this 
450  kw.  5900-  volt  three-phase  machine? 


18       ELEMENTS  OF  ELECTRICAL  ENGINEERING 


450  kw.  at  5900  volts  gives  44.2  amp.  per  line.  44.2  amp. 
effective  gives  44.2\/2  =  62.5  amp.  maximum. 

14  miles  =  22,400  m.  The  magnetic  flux  produced  by  / 
amperes  in  1000  m.  of  a  transmission  line  of  2  wires  45  cm. 
apart  and  0.82  cm.  diameter  was  found  in  paragraph  1,  example 
3,  as  2  $  =  0.188  X  106/,  or  $  =  0.094  X  106/  for  each  wire. 

Thus  at  22,300  m.  and  62.5  amp.  maximum,  the  flux  per 
wire  is 

$  =  22.3  X  62.5  X  0.094  X  106  =  131  megalines. 
Hence  the  generated  e.m.f.,  effective  value,  at  33.3  cycles  is, 

E  =  A/2  */  $ 

=  4.44  X  0.333  X  131 
=  193  volts  per  line; 
the  maximum  value  is, 

#o  =  E  X  \/2  =  273  volts  per  line; 
and  the  instantaneous  value, 

e  =  E0  sin   (0  -  0i)  =  273   sin   (0  -  0i) ; 

or,  since  0  =  2  irft  =  210  t  we  have, 

e  =  273  sin  210  (t  -  h). 

25.  (4)  What  is  the  form 
factor  (a)  of  the  e.m.f.  gene- 
rated in  a  single  conductor  of 
a  direct-current  machine  hav- 
ing 80  per  cent,  pole  arc  and 
negligible  spread  of  the  mag- 
netic flux  at  the  pole  corners, 
and  (6)  what  is  the  form  fac- 
tor of  the  voltage  between  two 
collector  rings  connected  to 
diametrical  points  of  the  arm- 
ature of  such  a  machine? 

(a)  In  a  conductor  during 
the  motion  from  position  A, 

T?     *,     T^-  shown  in  Fig.  7,  to  position 

FIG.  7. — Diagram  of  bipolar  generator. 

15,    no    e.m.f.    is    generated; 

from  position  B  to  C  a  constant  e.m.f.  e  is  generated,  from 
C  to  E  again  no  e.m.f.,  from  E  to  F  a  constant  e.m.f.  —  e, 


POWER  AND  EFFECTIVE  VALUES 


19 


and  from  F  to  A  again  zero  e.m.f.     The  e.m.f.  wave  thus  is  as 
shown  in  Fig.  8. 
The  average  e.m.f.  is 

ei  =  0.8  e; 

hence,  with  this  average  e.m.f.,  if  it  were  a  sine  wave,  the  maxi- 
mum e.m.f.  would  be 

ez  =  |  ei  =  0.4  ire, 
and  the  effective  e.m.f.  would  be 


C    D 


FIG.  8. — E.m.f.  of  a  single  conductor,  direct-current  machine 
80  per  cent,  pole  arc. 

The  actual  square  of  the  e.m.f.  is  e2  for  80  per  cent,  and  zero 
for  20  per  cent,  of  the  period,  and  the  average  or  mean  square 
thus  is 

0.8  e2, 

and  therefore  the  actual  effective  value, 


The  form  factor  7,  or  the  ratio  of  the  actual  effective  value 
e4  to  the  effective  value  e3  of  a  sine  wave  of  the  same  mean 
value  and  thus  the  same  magnetic  flux,  then  is 

e4      VT6 
T  =  e3  =  ^T 

=  1.006; 
that  is,  practically  unity. 

(6)  While  the  collector  leads  a,  b  move  from  the  position  F, 
C,  as  shown  in  Fig.  6,  to  B,  E,  constant  voltage  E  exists  between 
them,  the  conductors  which  leave  the  field  at  C  being  replaced 


20       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

by  the  conductors  entering  the  field  at  B.  During  the  motion 
of  the  leads  a,  b  from  B,  E  to  C,  F,  the  voltage  steadily  decreases, 
reverses,  and  rises  again,  to  —  E,  as  the  conductors  entering  the 
field  at  E  have  an  e.m.f.  opposite  to  that  of  the  conductors 
leaving  at  C.  Thus  the  voltage  wave  is,  as  shown  by  Fig.  9, 
triangular,  with  the  top  cut  off  for  20  per  cent,  of  the  half  wave. 


FIG.  9.  —  E.m.f.  between  two  collector  rings  connected  to  diametrical  points 
of  the  armature  of  a  bipolar  machine  having  80  per  cent,  pole  arc. 


Then  the  average  e.m.f.  is 

e1  =  0.2  E  +  2  X 


=  0.6  E. 


The  maximum  value  of  a  sine  wave  of  this  average  value  is 
e2  =  2  ei  —  0-3  irE, 

and  the  effective  value  corresponding  thereto  is 

e-2        0.3  irE 


63   = 


7= 

V2 


The  actual  voltage  square  is  E2  for  20  per  cent,  of  the  time,  and 
rising  on  a  parabolic  curve  from  0  to  E2  during  40  per  cent,  of 
the  time,  as  shown  in  dotted  lines  in  Fig.  9. 

The  area  of  a  parabolic  curve  is  width  times  one-third  of 
height,  or 

OAE2 


hence,  the  mean  square  of  voltage  is 

0 


and  the  actual  effective  voltage  is 


_,4       1/280 

~  e,  ~  TT  V  27        L°25' 


SELF-INDUCTANCE  AND  MUTUAL  INDUCTANCE    21 

hence,  the  form  factor  is 

7    r 

or,  2.5  per  cent,  higher  than  with  a  sine  wave. 

5.  SELF-INDUCTANCE  AND  MUTUAL  INDUCTANCE 

26.  The  number  of  inter-linkages  of  an  electric  circuit  with 
the  lines  of  magnetic  force  of  the  flux  produced  by  unit  current 
in  the  circuit  is  called  the  inductance  of  the  circuit. 

The  number  of  interlinkages  of  an  electric  circuit  with  the 
lines  of  magnetic  force  of  the  flux  produced  by  unit  current  in 
a  second  electric  circuit  is  called  the  mutual  inductance  of  the 
second  upon  the  first  circuit.  It  is  equal  to  the  mutual  induc- 
tance of  the  first  upon  the  second  circuit,  as  will  be  seen,  and 
thus  is  called  the  mutual  inductance  between  the  two  circuits. 

The  number  of  interlinkages  of  an  electric  circuit  with  the 
lines  of  magnetic  flux  produced  by  unit  current  in  this  circuit 
and  not  interlinked  with  a  second  circuit  is  called  the  self- 
inductance  of  the  circuit. 

If  i  =  current  in  a  circuit  of  n  turns,  <l>  =  flux  produced 
thereby  and  interlinked  with  the  circuit,  n$  is  the  total  number 

9?  ^^ 

of  interlinkages,  and  L  =  — r-  the  inductance  of  the  circuit. 
If  $  is  proportional  to  the  current  i  and  the  number  of  turns  n, 

ni  n2          .    , 

$  =  — ,  and  L  =  —  the  inductance. 
01  (K 

(ft  is  called  the  reluctance  and  ni  the  m.m.f.  of  the  magnetic 
circuit. 

In  magnetic  circuits  the  reluctance  (R  has  a  position  similar 
to  that  of  resistance  r  in  electric  circuits. 

The  reluctance  (R,  and  therefore  the  inductance,  is  not  con- 
stant in  circuits  containing  magnetic  materials,  such  as  iron,  etc. 

If  (Ri  is  the  reluctance  of  a  magnetic  circuit  interlinked  with 
two  electric  circuits  of  n\  and  n%  turns  respectively,  the  flux 
produced  by  unit  current  in  the  first  circuit  and  interlinked  with 

the  second  circuit  is  --  and  the  mutual  inductance  of  the  first 
(HI 

upon  the  second   circuit  is  M  = ,   that   is,    equal    to    the 

Oil 


22       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

mutual  inductance  of  the  second  circuit  upon  the  first  circuit, 
as  stated  above. 

If  no  flux  leaks  between  the  two  circuits,  that  is,  if  all  flux  is 
interlinked  with  both  circuits,  and  LI  =  inductance  of  the  first, 
L2  =  inductance  of  the  second  circuit,  and  M  =  mutual  induc- 
tance, then 

M2  =  ZaL2. 

If  flux  leaks  between  the  two  circuits,  then  M 2  <  LiL2. 

In  this  case  the  total  flux  produced  by  the  first  circuit  con- 
sists of  a  part  interlinked  with  the  second  circuit  also,  the  mu- 
tual inductance,  and  a  part  passing  between  the  two  circuits, 
that  is,  interlinked  with  the  first  circuit  only,  its  self-inductance. 

27.  Thus,  if  LI  and  L2  are  the  inductances  of  the  two  circuits, 

—  and  —  is  the  total  flux  produced  by  unit  current  in  the  first 
n\          HZ 

and  second  circuit  respectively. 

T  Sf 

Of  the  flux  --a  part  —  is  interlinked  with  the  first  circuit 
ni  HI 

only,  Si  being  its  self-inductance  or  leakage  inductance,  and  a 
part  —  interlinked  with  the  second  circuit  also,   M  being  the 

mutual  inductance  and  — 1  =  —  +  — 

HI       HI       n2 

Thus,  if 

LI  and  L2  =  inductance, 
Si  and  Sz  =  self-inductance, 

M  =  mutual  inductance    of   two    circuits  of  n  and 
n2  turns  respectively,  we  have 

h  =  Sl  +  *L  Lz  =  Sz      M 

HI       HI        nz  nz  ~  HZ        n\ 

or  Li  =  Si  +  —  M  L2  =  Sz  +  -  M, 

HZ  HI 

or  M2  =  (Li  -  Si)(Lz  -  Sz). 

The  practical  unit  of  inductance  is  109  times  the  absolute 
unit  or  108  times  the  number  of  interlinkages  per  ampere  (since 
1  amp.  =  0.1  unit  current),  and  is  called  the  henry  (h);  0.001 
of  it  is  called  the  milhenry  (mh.). 

The  number  of  interlinkages  of  i  amperes  in  a  circuit  of 


SELF-INDUCTANCE  AND  MUTUAL  INDUCTANCE    23 

L  henry  inductance  is  iL  108  lines  of  force  turns,  and  thus  the 
e.m.f.  generated  by  a  change  of  current  di  in  time  dt  is 

e  =  —  -j-.  L  108  absolute  units 

r  dt 

=  —  -T.L  volts. 
at 

A  change  of  current  of  1  amp.  per  second  in  the  circuit  of  1 
h.  inductance  generates  1  volt. 


EXAMPLES 

28.  (1)  What  is  the  inductance  of  the  field  of  a  20-pole 
alternator,  if  the  20  field  spools  are  connected  in  series,  each 
spool  contains  616  turns,  and  6.95  amp.  produces  6.4  mega- 
lines  per  pole? 

The  total  number  of  turns  of  all  20  spools  is  20  X  616  = 
12,320  Each  is  interlinked  with  6.4  X  106  lines,  thus  the  total 
number  of  interlinkages  at  6.95  amp.  is  12,320  X  6.4  X  106  = 
78  X  109. 

6.95  amp.  =  0.695  absolute  units,  hence  the  number  of  in- 
terlinkages per  unit  current,  or  the  inductance,  is 

-  112  X  I*-  112  h. 


29.  (2)  What  is  the  mutual  inductance  between  an  alter- 
nating transmission  line  and  a  telephone  wire  carried  for  10  miles 
below  and  1.20  m.  distant  from  the  one,  1.50  m.  distant  from 
the  other  conductor  of  the  alternating  line;  and  what  is  the 
e.m.f.  generated  in  the  telephone  wire,  if  the  alternating  cir- 
cuit carries  100  amp.  at  60  cycles? 

The  mutual  inductance  between  the  telephone  wire  and  the 
electric  circuit  is  the  magnetic  flux  produced  by  unit  current 
in  the  telephone  wire  and  interlinked  with  the  alternating 
circuit,  that  is,  that  part  of  the  magnetic  flux  produced  by  unit 
current  in  the  telephone  wire,  which  passes  between  the  dis- 
tances of  1.20  and  1.50  m. 

At  the  distance  lx  from  the  telephone  wire  the  length  of  mag- 

netic circuit  is  2irlz.     The  magnetizing  force  /  =     —-  if  7  = 


24       ELEMENTS  OF  ELECTRICAL  ENGINEERING 


current  in  telephone  wire  in  amperes,  and  the  field   intensity 
d  the 


0.27 

H  =  0.4  TT/  =  —  —  ,  and  the  flux  in  the  zone  dlx  is 


j       dlx. 
lx 

I  =  10  miles  =  1610  X  103  cm. 


thus, 


f1500.2// 
=  I        —  i  —  dlx 

Jl20          *» 


=  322  X  10371ogei||°  =  72  7  103; 


or,     72  7  103  interlinkages,  hence,  for  7  =  10,  or  one  absolute 

unit, 

thus,  M  =  72  X  104  absolute  units  =  72  X  10~5  h.  =  0.72  mh. 
100  amp.  effective  or  141.4  amp.  maximum  or  14.14  abso- 
lute units  of  current  in  the  transmission  line  produces  a 
maximum  flux  interlinked  with  the  telephone  line  of  14.14  X 
0.72  X  10~3  X  109  =  10.2  megalines.  Thus  the  e.m.f.  generated 
at  60  cycles  is 

E  =  4.44  X  0.6  X  10.2  =  27.3  volts  effective. 

6.   SELF-INDUCTANCE  OF  CONTINUOUS-CURRENT 
CIRCUITS 

30.  Self-inductance  makes  itself  felt  in  continuous-current 
circuits  only  in  starting  and  stopping  or,  in  general,  when  the 
current  changes  in  value. 

Starting  of  Current.  If  r  =  resistance,  L  =  inductance  of 
circuit,  E  =  continuous  e.m.-f.  impressed  upon  circuit,  i  = 
current  in  circuit  at  time  t  after  impressing  e.m.f.  E,  and  di  the 
increase  of  current  during  time  moment  dt,  then  the  increase  of 
magnetic  interlinkages  during  time  dt  is 

IM, 
and  the  e.m.f.  generated  thereby  is 

r  di 

ei  =  -L~di 

By  Lentz's  law  it  is  negative,  since  it  is  opposite  to  the   im- 
pressed e.m.f.,  its  cause. 

Thus  the  e.m.f.  acting  in  this  moment  upon  the  circuit  is 

E  +  ei  =  E  -  L  § 


CONTINUOUS-CURRENT  CIRCUITS  25 


and  the  current  is 


or,  transposing, 


™  _  r 

dt 


rdt          di 


L        i- 

r 

the  integral  of  which  is 

.       E 


rt       .        /.       E\      . 
-  L  --=  log,  [i  -  -)  -  log,  c, 


where  —  log,  c  =  integration  constant. 
This  reduces  to    - 

•       E  _i        -  7 
i  =  -  +  ce      L 

at  t  =  0,  i  =  0,  and  thus 

E 

-~  =  c. 

Substituting  this  value,  the  current  is 


and  the  e.m.f.  of  inductance  is 

Att   =    co  , 

i'o 
Substituting  these  values, 


_  rf 

=  ir  —  E  =  —  Ee~  L' 


r» 

-'        «i  =  0. 


i  =  iQ  (l  -  e 
and 


Z. 
The  expression  u  =  j  is  called  the  "attenuation  constant," 

and  its  reciprocal,  —  ,  the  "time  constant  of  the  circuit."1 

1  The  name  time  constant  dates  back  to  the  early  days  of  telegraphy,  where 
it  was  applied  to  the  ratio  :  —  ,  that  is,  the  reciprocal  of  the  attenuation  con- 

stant. This  quantity  which  had  gradually  come  into  disuse,  again 
became  of  importance  when  investigating  transient  electric  phenomena, 

and  in  this  work  it  was  found  more  convenient  to  denote  the  value  Y  as 
attenuation  constant,  since  this  value  appears  as  one  term  of  the  more  gen- 
eral constant  of  the  electric  circuit  (  Y  +  ~r<  )  •  (Theory  and  Calculation  of 
Transient  Electric  Phenomena  and  Oscillations,  Section  IV.) 


26       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Substituted  in  the  foregoing  equation  this  gives 


and 


ei  =  -      =  -  0.368  E. 

31.  Stopping  of  Current.     In  a  circuit  of  inductance  L  and 

E 

resistance  r,  let  a  current  IQ  =  —  be  produced  by  the  impressed 

e.m.f.  E,  and  this  e.m.f.  E  be  withdrawn  and  the  circuit  closed 
through  a  resistance  r\. 

Let  the  current  be  i  at  the  time  t  after  withdrawal  of  the 
e.m.f.  E  and  the  change  of  current  during  time  moment  dt  be  di. 
di  is  negative,  that  is,  the  current  decreases. 

The  decrease  of  magnetic  interlinkages  during  moment  dt  is 

Ldi. 

Thus  the  e.m.f.  generated  thereby  is 

Tdi 

ei==  ~Ldi 

It  is  negative  since  di  is  negative,  and  e\  must  be  positive,  that 
is,  in  the  same  direction  as  E,  to  maintain  the  current  or  oppose 
the  decrease  of  current,  its  cause. 
Then  the  current  is 

e\  L     di 


_ 

=  r  +  n  r  +  ri  dt 


or,  transposing, 


the  integral  of  which  is 


t  =  log,  i  -  log,  c, 


where—  log€  c  =  integration  constant. 

r+n 

This  reduces  to  i  =  ce        L 

77* 
for  t  =  0,        t 


CONTINUOUS-CURRENT  CIRCUITS  27 

Substituting  this  value,  the  current  is 


E          (r  +  n)  t 
i  =  -€~         L        , 

and  the  generated  e.m.f.  is 

r  +  r,   _fc  + 
ei  =•  t  (r  +  fi)  •»  Js  —    —  c        i 

r 

•p 

Substituting  IQ  =  — ,  the  current  is 


i  =  lot 
and  the  generated  e.m.f.  is 


6l  =  H(r-f  TI)€       JT-', 

At  *  =  0, 


that  is,  the  generated  e.m.f.  is  increased  over  the  previously 
impressed  e.m.f.  in  the  same  ratio  as  the  resistance  is  increased. 
When  TI  =  0,  that  is,  when  in  withdrawing  the  impressed 
e.m.f.  E  the  circuit  is  short  circuited, 

E  _L«          _';! 
i  =  —  c    L  =  io€    z,  the  current,  and 

_  TL  _  !l 

ei  =  E  €    L  =  i0re    £  the  generated  e.m.f. 


In  this  case,  at  t  =  0,  e\  =  E,  that  is,  the  e.m.f.  does  not  rise. 

In  the  case  r\  =  <»  ,  that  is,  if  in  withdrawing  the  e.m.f.  E 
the  circuit  is  broken,  we  have  t  =  0  and  ei  =  »  ,  that  is,  the 
e.m.f.  rises  infinitely. 

The  greater  r\,  the  higher  is  the  generated  e.m.f.  e\,  the  faster, 
however,  do  e\  and  i  decrease. 

If  n  =  r,  we  have  at  t  =  0, 

611  =  2E,  i  =  i0, 

and  en  —  i<jr  =  #; 

that  is,  if  the  external  resistance  7*1  equals  the  internal  resistance 
r,  at  the  moment  of  withdrawal  of  the  e.m.f.  E  the  terminal 
voltage  is  E. 


28       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

The  effect  at  the  time  t  of  the  e.m.f.  of  inductance  in  stop- 
ping the  current  is 

_  2r  +  rif 

iei  =  io2  (r  +  n)  c         L      ; 
thus  the  total  energy  of  the  generated  e.m.f. 

>*» 

W  =  |      z' 

Jo 


that  is,  the  energy  stored  as  magnetism  in  a  circuit  of  current  iQ 
and  inductance  L  is 


2  ' 

which  is  independent  both  of  the  resistance  r  of  the  circuit  and 
the  resistance  n  inserted  in  breaking  the  circuit.  This  energy 
has  to  be  expended  in  stopping  the  current. 

EXAMPLES 

32.  (1)  In  the  alternator  field  in  Section  1,  Example  4,  Sec- 
tion 2,  Example  2,  and  Section  5,  Example  1,  how  long  a  time 
after  impressing  the  required  e.m.f.  E  =  230  volts  will  it  take 
for  the  field  to  reach  (a)  J/£  strength,  (b)  %Q  strength? 

(2)  If  500  volts  are  impressed  upon  the  field  of  this  alternator, 
and  a  non-inductive  resistance  inserted  in  series  so  as  to  give 
the  required   exciting   current   of  6.95   amp.,    how  long  after 
impressing  the  e.m.f.  E  =  500  volts  will  it  take  for  the  field  to 
reach  (a)  y%  strength,  (b)  %o  strength,  (c)  and  what  is  the  resist- 
ance required  in  the  rheostat? 

(3)  If  500  volts  are  impressed  upon  the  field  of  this  alter- 
nator without  insertion  of  resistance,  how  long  will  it  take  for 
the  field  to  reach  full  strength? 

(4)  With  full  field   strength,   what  is  the  energy  stored   as 
magnetism? 

(1)  The  resistance  of  the  alternator  field  is  33.2  ohms  (Section 
2,  Example  2),  the  inductance  112  h.  (Section  5,  Example  1), 
the  impressed  e.m.f.  is  E  =  230,  the  final  value  of  current 

E 

io  =  —  =  6.95  amp.     Thus  the  current  at  time  t  is 


t  =  *        -  6 
=  6.95  (1  -  e-°-296<). 


CONTINUOUS-CURRENT  CIRCUITS  29 

(a)  M  strength:  i  =  ~,  hence  (1  -  €-°-29«0  =  0.5. 
e-o.296  1  =  0.5,   -  0.296  Mog  e  =  log  0.5,    t  = 
t  =  2.34  seconds. 


(b)  %0  strength:  i  =  0.9  *0j  hence  (1  -  e-°-2960  =  0.9,  and 
t  =  7.8  seconds. 

(2)  To   get   io  =  6.95   amp.,  with  E  =  500   volts,    a   resist- 

500 

ance  r  =  ^-f-=  =  72  ohms,  and  thus  a  rheostat  having  a  resist- 
o.9o 

ance  of  72  —  33.2  =  38.8  ohms,  is  required. 
We  then  have 

i    =   io   (l  €     2) 

=  6.95  (1  -  €-°-6430. 

(a)  i  =  ^,  after  t  =  1.08  seconds. 

& 

(b)  i  =  0.9  i0,  after  i  =  3.6  seconds. 

(3)  Impressing  E  =  500  volts    upon    a  circuit  of  r  =  33.2, 
L  =  112,  gives 


=  15.1  (1  -  €-°-2960. 
i  =  6.95,  or  full  field  strength,  gives 
6.95  =  15.1  (1  -  e-°-2960. 

1  -  €-°-296  «  =  0.46 

and  t  =  2.08  seconds. 

(4)  The  stored  energy  is 
6.952  X  112 


~  =  2720  watt-seconds  or  joules 

_  _ 

=  2000  foot-pounds. 

(1  joule  =  0.736  foot-pounds.) 
Thus  in  case  (3),  where  the  field  reaches  full  strength  in  2.08 

2000 
seconds,  the  average  power  input  is  -c  -^  =  960  foot  pounds 

Z.Oo 

per  second  =  1.75  hp. 

In  breaking  the  field  circuit  of  this  alternator,  2000  foot- 
pounds of  energy  have  to  be  dissipated  in  the  spark,  etc. 

33.  (5)  A  coil  of  resistance  r  =  0.002  ohm  and  inductance 
L  =  0.005  mh.,  carrying  current  /  =  90  amp.,  is  short  circuited. 


30       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

(a)  What  is  the  equation  of  the  current  after  short  circuit? 
(6)  In  what  time  has  the  current  decreased  to  0.1.  its  initial 
value? 

_  L* 

(a)  i  =  /e    L 

=    90  e-400'. 

(6)  i  =  0.1  7,  c-400<  =  0.1,  after  t  =  0.00576  second. 

(6)  When  short  circuiting  the  coil  in  Example  5,  an  e.m.f. 
E  =  1  volt  is  inserted  in  the  circuit  of  this  coil,  in  opposite  direc- 
tion to  the  current. 

(a)  What  is  equation  of  the  current? 

(6)  After  what  time  does  the  current  become  zero? 

(c)  After  what  time  does  the   current  reverse  to  its  initial 
value  in  opposite  direction? 

(d)  What  impressed  e.m.f.  is  required  to  make  the  current 
die  out  in  Hooo  second? 

(e)  What  impressed  e.m.f.  E  is  required  to  reverse  the  current 
in  Kooo  second? 

(a)  If  e.m.f.  —  E  is  inserted,  and  at  time  t  the  current  is 
denoted  by  i,  we  have 

di 
ei  =  —  L  -r,  the  generated  e.m.f. ; 

Thus,  -  E  +  61  =  -  E  -  L  jt,  the  total  e.m.f.; 

and 

-E  +  ei  E      L  di  ^ 

i  = = -r.,  the  current; 

r  r        r  dt} 

Transposing, 

r  ,  di 


and  integrating, 

-  j-  =  log,  (-  +  i)  -  log,  c, 


where  —  log^  c  =  integration  constant. 

At  t  =  0,  i 
Substituting, 


E 

At  t  =  0,  i  =  /,  thus  c  =  /  +  -; 


t  E\    -«      E 

h7r      "7' 

Kon  ,—400 1  p;nn 

*J*S\J    C  <J\J\Jt 


ALTERNATING-CURRENT  CIRCUITS  31 

(6)  i  =  o,  e-400  «  =  0.85,  after  t  =  0.000405  second. 

(c)  i  =  -  I  =  -  90,  e-400  «  =  0.694,  after  t  =  0.00091  second. 

(d)  If  i  =  0  at  *  =  0.0005,  then 

0  =  (90  +  500  E)  e-°-2  -  500  #, 

E  =  joTITi  =  °-81  volt- 

(e)  If  t  =  -  I  =  -  90  at  t  =  0.001,  then 
-  90  =  (90  +  500  E)  €-°-4  -  500  E, 


E  =     -  =  0.91  volt. 


7.  INDUCTANCE  IN  ALTERNATING-CURRENT  CIRCUITS 

34.  An  alternating  current  i  =  IQ  sin  2irft  or  i  —  I0  sin  0 
can  be  represented  graphically  in  rectangular  coordinates  by  a 
curved  line  as  shown  in  Fig.  10,  with  the  instantaneous  values 


FIG.  10. — Alternating  sine  wave. 

i  as  ordinates  and  the  time  t,  or  the  arc  of  the  angle  corresponding 
to  the  time,  6  =  2irft,  as  abscissas,  counting  the  time  from  the 
zero  value  of  the  rising  wave  as  zero  point. 

If  the  zero  value  of  current  is  not  chosen  as  zero  point  of  time, 
the  wave  is  represented  by 

i  =  /0  sin  2  IT/ (t  -  t'), 
or  i  =  /osin  (6  —  8'), 

where  tf  and  6'  are  respectively  the  time  and  the  corresponding 
angle  at  which  the  current  reaches  its  zero  value  in  the  ascendant. 
If  such  a  sine  wave  of  alternating  current  i  =  IQ  sin  2  irft  or 
i  =  IQ  sin  6  passes  through  a  circuit  of  resistance  r  and  induc- 
tance L,  the  magnetic  flux  produced  by  the  current  and  thus  its 
interlinkages  with  the  current,  iL  =  IoL  sin  0,  vary  in  a  wave 


32       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

line  similar  also  to  that  of  the  current,  as  shown  in  Fig.  11  as  $. 
The  e.m.f.  generated  hereby  is  proportional  to  the  change  of 
iL,  and  is  thus  a  maximum  where  iL  changes  most  rapidly,  or  at 
its  zero  point,  and  zero  where  iL  is  a  maximum,  and  according 
to  Lentz's  law  it  is  positive  during  falling  and  negative  during 
rising  current.  Thus  this  generated  e.m.f.  is  a  wave  following 

the  wave  of  current  by  the  time  t  =  -,>    where  tQ  is  time  of  one 

1 
complete  period,  =  -v    or  by  the  time  angle  6  =  90°. 


FIG.  11. — Self-induction   effects   produced   by   an   alternating   sine   wave 

of  current. 

This  e.m.f.  is  called  the  counter  e.m.f.  of  inductance.     It  is 

.'•'•     '••••     e'*=-Ljt 

=  -  2  TT/L/O  cos  2  irft. 

It  is  shown  in  dotted  line  in  Fig.  11  as  e'2. 

The  quantity  2  irfL  is  called  the  inductive  reactance  of  the 
circuit,  and  denoted  by  x.  It  is  of  the  nature  of  a  resistance, 
and  expressed  in  ohms.  If  L  is  given  in  109  absolute  units  or 
henrys,  x  appears  in  ohms. 

The  counter  e.m.f.  of  inductance  of  the  current,  i  =  /o  sin 
2  irft  =  /o  sin  0)  of  effective  value 


, 

V"2 


IS 


e'2  =  —  xI0  cos  2  irft  =  —  xIQ  cos  6, 
having  a  maximum  value  of  X!Q  and  an  effective  value  of 

xh          T 
E,  =      --  =  xl; 


ALTERNATING-CURRENT  CIRCUITS  33 

that  is,  the  effective  value  of  the  counter  e.m.f.  of  inductance 
equals  the  reactance,  x,  times  the  effective  value  of  the  current, 
/,  and  lags  90  time  degrees,  or  a  quarter  period,  behind  the 
current. 

35.  By  the  counter  e.m.f.  of  inductance, 

e'z  =  —  xIQ  cos  0, 

which  is  generated  by  the  change  in  flux  due  to  the  passage  of 
the  current  i  —  IQ  sin  0  through  the  circuit  of  reactance  x,  an 
equal  but  opposite  e.m.f. 

ez  =  xIQ  cos  0 

is  consumed,  and  thus  has  to  be  impressed  upon  the  circuit. 
This  e.m.f.  is  called  the  e.m.f.  consumed  by  inductance.  It  is 
90  time  degrees,  or  a  quarter  period,  ahead  of  the  current,  and 
shown  in  Fig.  11  as  a  drawn  line  e2. 

Thus  we  have  to  distinguish  between  counter  e.m.f.  of  induc- 
tance 90  time  degrees  lagging,  and  e.m.f.  consumed  by  inductance 
90  time  degrees  leading. 

These  e.m.fs.  stand  in  the  same  relation  as  action  and  reaction 
in  mechanics.  They  are  shown  in  Fig.  11  as  e'2  and  as  ez. 

The  e.m.f.  consumed  by  the  resistance  r  of  the  circuit  is  pro- 
portional to  the  current, 

61  =  ri  =  r/0  sin  0, 

and  in  phase  therewith,  that  is,  reaches  its  maximum  and  its 
zero  value  at  the  same  time  as  the  current  i,  as  shown  by  drawn 
line  61  in  Fig.  11. 

Its  effective  value  is  EI  =  ri. 

The  resistance  can  also  be  represented  by  a  (fictitious)  counter 
e.m.f., 

e\  =  —  r/0  sin  0, 

opposite  in  phase  to  the  current,  shown  as  e\  in  dotted  line  in 
Fig.  11. 

The  counter  e.m.f.  of  resistance  and  the  e.m.f.  consumed  by 
resistance  have  the  same  relation  to  each  other  as  the  counter 
e.m.f.  of  inductance  and  the  e.m.f.  consumed  by  inductance  or 
inductive  reactance. 

36.  If  an  alternating  current  i  =  70  sin  0  of  effective  value 

I  =  —^  exists  in  a  circuit  of  resistance  r  and  inductance  L,  that 
is,  of  reactance  x  =  2  irfL,  we  have  to  distinguish ; 

3 


34       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

E.m.f.  consumed  by  resistance,  e\  =  r!0  sin  6,  of  effective 
value  EI  =  rl,  and  in  phase  with  the  current. 

Counter  e.m.f.  of  resistance,  e'\  =  —  r/o  sin  6,  of  effective 
value  EI  =  rl,  and  in  opposition  or  180  time  degrees  displaced 
from  the  current. 

E.m.f.  consumed  by  reactance,  ez  =  X!Q  cos  6,  of  effective 
value  E2  =  xl,  and  leading  the  current  by  90  time  degrees  or  a 
quarter  period. 

Counter  e.m.f.  of  reactance,  e'z  =  xlo  cos  0,  of  effective 
value  E'z  =  xl,  and  lagging  90  time  degrees  or  a  quarter  period 
behind  the  current. 

The  e.m.fs.  consumed  by  resistance  and  by  reactance  are  the 
e.m.fs.  which  have  to  be  impressed  upon  the  circuit  to  overcome 
the  counter  e.m.fs.  of  resistance  and  of  reactance. 

Thus,  the  total  counter  e.m.f.  of  the  circuit  is 

e'  =  e'i  +  e'z  =  —  IQ  (r  sin  6  +  x  cos-0), 

and  the  total  impressed  e.m.f.,  or  e.m.f.  consumed  by  the  circuit, 
is 

e  =  ei  +  ez  =  /o  (r  sin  6  +  x  cos  0). 
Substituting 

9S 

-  =  tan  00  and 

VV2  +  r2  =  z, 
it  follows  that 

x  =  z  sin  00,  r  =  z  cos  00, 

and  we  have  as  the  total  impressed  e.m.f. 

e  =  Z!Q  sin  (0  +  00), 
shown  by  heavy  drawn  line  e  in  Fig.  11,  and  total  counter  e.m.f. 

e'  =  -  zI0  sin  (0  +  00), 

shown  by  heavy  dotted  line  e'  in  Fig.  11,  both  of  effective  value 

e  =  zl. 

For  0  =  —  00,  e  —  0,  that  is,  the  zero  value  of  e  is  ahead  of 
the  zero  value  of  current  by  the  time  angle  0o,  or  the  current  lags 
behind  the  impressed  e.m.f.  by  the  angle  00. 

0o  is  called  the  angle  of  lag  of  the  current,  and  z  =  \A"2  +  x2 
the  impedance  of  the  circuit,  e  is  called  the  e.m.f.  consumed  by 
impedance,  e'  the  counter  e.m.f.  of  impedance. 


ALTERNATING-CURRENT  CIRCUITS  35 

Since  Ei  =  rl  is  the  e.m.f.  consumed  by  resistance, 
Ez  =  xl  is  the  e.m.f.  consumed  by  reactance, 

and          E  =  zl  =  \/r2  +  x2  1  is  the  e.m.f.  consumed  by  impe- 

dance, 
we  have 


E   =  VES  +  #22,  the  total  e.m.f. 

and  Ei  =  E  cos  00, 

E%  =  E  sin  0o,  its  components. 

The  tangent  of  the  angle  of  lag  is 

x       27T/L 
tan  00  =  -  =  —  —  > 

and  the  time  constant  of  the  circuit  is 

L  _  tan  &o 
r  =    ~27f" 

The  total  e.m.f.,  e,  impressed  upon  the  circuit  consists  of  two 
components,  one,  e\t  in  phase  with  the  current,  the  other  one,  e2, 
in  quadrature  with  the  current. 

Their  effective  values  are 

E,  E  cos  00)  E  sin  00. 

EXAMPLES 

37.  (1)  What  is  the  reactance  per  wire  of  a  transmission  line 
of  length  Z,  if  ld  =  diameter  of  the  wire,  18  =  spacing  of  the  wires, 
and/  =  frequency? 

If  /  =  current,  in  absolute  units,  in  one  wire  of  the  trans- 
mission line,  the  m.m.f.  is  I;  thus  the  magnetizing  force  in  a 

zone  dlx  at  distance  lx  from  center  of  wire  (Fig.  12)  is  /  =  0    7 

Z  TTlx 

and  the  field  intensity  in  this  zone  is  H  =  4  irf  =  2  y—     Thus 

Lx 

the  magnetic  flux  in  this  zone  is 

d*  .  H  ldli  m 


hence,  the  total  magnetic  flux  between  the  wire  and  the  return 
wire  is 

L 


XI* 
d*  = 


$  —      |        CfcSF  =  ^.f6|         -y—    =    2  1 1  IQge  -j — > 

LX  I'd 

2  2 

neglecting  the  flux  inside  the  transmission  wire. 


36       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

The  inductance  is 

L  =  -y  =  2  I  loge  -y^  absolute  units 

-/  'd 


2  Z  loge      s  1(T9  h., 

I'd 


21 
s 


and  the  reactance  x  =  2  irfL  =  4  irfl  loge  -y-,  in  absolute  units; 
or  x  =  4  7T/7  loge  -y^  10~9,  in  ohms. 

'd 

38.  (2)    The    voltage   at  the  receiving   end   of   a  33.3-cycle 
three-phase  transmission  line  14  miles  in  length  shall  be  5500 


FIG.  12. — Diagram    for    calculation    of  inductance  between  two  parallel 

conductors. 

between  the  lines.  The  line  consists  of  three  wires,  No.  0  B.  & 
S.  (ld  =  0.82  cm.),  18  in.  (45  cm.)  apart,  of  resistivity  p  =  1.8 
X  10-6. 

(a)  What  is  the  resistance,  the  reactance,  and  the  impedance 
per  line,  and  the  voltage  consumed  thereby  at  44  amp.  ? 

(6)  What  is  the  generator  voltage  between  lines  at  44  amp. 
to  a  non-inductive  load? 

(c)  What  is  the  generator  voltage  between  lines  at  44  amp. 
to  a  load  circuit  of  45  degrees  lag? 

(d)  What  is  the  generator  voltage  between  lines  at  44  amp. 
to  a  load  circuit  of  45  degrees  lead? 

Here  I  =  14  miles  =  14  X  1.6  X  105  =  2.23  X  106  cm. 
ld  =  0.82  cm. 

Hence  the  cross  section,  A  =  0.528  sq.  cm. 


ALTERNATING-CURRENT  CIRCUITS  37 

Z        1.8  X  10-6  X  2.23  X  106 
(a)  Resistance    per    line,    r  =  p  -   = 


=  7.60  ohms. 

2L 
Reactance  per  line,  x  =  4  irfl  log,  j-  X  10~9  =  ^  X  33.3  X 

2.23  X  106  X  log,  110  X  10~9  =  4.35  ohms. 

The  impedance  per  line,  z  =  \/r2  -f-  x'2  =  8.76  ohms.  Thus 
if  I  =  44  amp.  per  line, 

the  e.m.f.  consumed  by  resistance  is  EI  =  rl  =  334  volts, 
the  e.m.f.  consumed  by  reactance  is  Ez  =  xl  =  192  volts, 
and  the  e.m.f.  consumed  by  impedance  is  E3  =  zl  =  385  volts. 

(b)  5500  volts  between  lines  at  receiving  circuit  give  —  -j=  = 

v  3 

3170  volts  between  line  and  neutral  or  zero  point   (Fig.   13), 
or  per  line,  corresponding  to  a  maxi- 
mum voltage  of  3170  A/2  =  4500  volts. 
44  amp.  effective  per  line  gives  a  maxi- 
mum value  of  44  -\/2  =  62  amp. 

Denoting  the  current  by  i  =  62  sin 
0,  the  voltage  per  line  at  the  receiv- 
ing end  with  non-inductive  load  is  e 
=  4500  sin  0. 

The  e.m.f.  consumed  by  resistance, 
in  phase  with  the  current,  of  effective  JTIG  13  _  Voltage  diagram  for 
value   334,  and  maximum  value  334          a  three-phase  circuit. 
\/2  =  472,  is 

ei  =  472  sin  0. 

The  e.m.f.  consumed  by  reactance,  90  time  degrees  ahead  of  the 
current,  of  effective  value  192,  and  maximum  value  192  -\/2  = 
272,  is 

e2  =  272  cos  0. 

Thus  the  total  voltage  required  per  line  at  the  generator  end 
of  the  line  is 

eQ  =  e  +  el  +  e2=  (4500  +  472)  sin  0  +  272  cos  0 
=  4972  sin  0  +  272  cos  0. 

272 
Denoting   .n_0  =  tan  00,  we  have 

tan  0o  272 


C°S  "°  = 


4980 

1  _  4972 

~  4980- 


38       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Hence,  BQ  =  4980  (sin  0  cos  0o  +  cos  6  sin  00) 

=  4980  sin  (0  +  00). 

Thus  0o  is  the   lag  of   the  current  behind  the  e.m.f.  at  the 
generator  end  of  the  line,    =  3.2  time  degrees,  and  4980  the 

4980 
maximum  voltage  per  line  at  the  generator  end;  thus  EQ  = 


=  3520,  the  effective  voltage  per  line,  and  3520  \/3  =  6100,  the 
effective  voltage  between  the  lines  at  the  generator. 

(c)  If  the  current 

i  =  62  sin  0 

lags  in  time  45  degrees  behind  the  e.m.f.  at  the  receiving  end  of 
the  line,  this  e.m.f.  is  expressed  by 

e  =  4500  sin  (0  +  45)  =  3170  (sin  0  +  cos  0); 

that  is,  it  leads  the  current  by  45  time  degrees,  or  is  zero  at  0  = 
—  45  time  degrees. 

The  e.m.f.  consumed  by  resistance  and  by  reactance  being  the 
same  as  in  (6),  the  generator  voltage  per  line  is 

€o  =  e  -f  6l  -f  e2  =  3642  sin  0  +  3442  cos  0. 

3442 
Denoting  QA/fo  =  tan  00,  we  have 

OO4Z 

e0  =  5011  sin  (0  -f-  00). 

Thus  00,  the  angle  of  lag  of  the  current-  behind  the  gen- 
erator e.m.f.,  is  43  degrees,  and  5011  the  maximum  voltage; 
hence  3550  the  effective  voltage  per  line,  and  3550  -\/3  =  6160 
the  effective  voltage  between  lines  at  the  generator. 

(d)  If  the  current  i  =  62  sin  0  leads  the  e.m.f.  by  45  degrees, 
the  e.m.f.  at  the  receiving  end  is 

e  =  4500  sin  (0  -  45) 
=  3180  (sin  0  -  cos  0). 

Thus  at  the  generator  end 

€Q  =  e  -f  6l  +  e2  =  3652  sin  0  -  2908  cos  0. 

2908 
Denoting  =  tan  ^o,  it  is 


e0  =  4670  sin  (0  -  00). 

Thus  00,  the  time  angle  of  lead  at  the  generator,  is  39  degrees, 
and  4654  the  maximum  voltage;  hence  3290  the  effective  vol- 
tage per  line  and  5710  the  effective  voltage  between  lines  at  the 
generator. 


POWER  IN  ALTERNATING-CURRENT  CIRCUITS        39 


8.  POWER  IN  ALTERNATING-CURRENT  CIRCUITS 


of  effective  value  I  =  —7=-,  in  a  circuit  of  resistance  r  and  reac- 

V2 


39.  The  power  consumed  by  alternating  current  i  =  I0  sin  0, 
effective  value  I 

tance  x  =  2  nfL,  is 

p  =  ei, 

where  e  =  z!Q  sin  (0  +  00)  is  the  impressed  e.m.f.,  consisting  of 
the  components 

ei  =  r/0  sin  0,  the  e.m.f.  consumed  by  resistance 
and         62  =  x!Q  cos  0,  the  e.m.f.  consumed  by  reactance. 

z  =  \/r2  +  x2  is  the  impedance  and  tan  00  =  —  the  phase  angle 
of  the  circuit;  thus  the  power  is 

p  =  z/o2  sin  0  sin  (0  +  00) 

=  ^-  (€OS  00   -   COS  (20+   00)) 

=  zP  (cos  00  -  cos  (20  +  00)). 
Since  the  average  cos  (20  +  00)  =  zero,  the  average  power  is 

P  =  zP  cos  00 
=  rP  =  EJ-, 

that  is,  the  power  in  the  circuit  is  that  consumed  by  the  resistance, 
and  independent  of  the  reactance. 

Reactance  or  self-inductance  consumes  no  power,  and  the 
e.m.f.  of  self-inductance  is  a  wattless  or  reactive  e.m.f.,  while  the 
e.m.f.  of  resistance  is  a  power  or  active  e.m.f. 

The  wattless  e.m.f.  is  in  quadrature,  the  power  e.m.f.  in  phase 
with  the  current. 

In  general,  if  0  =  angle  of  time-phase  displacement  between 
the  resultant  e.m.f.  and  the  resultant  current  of  the  circuit, 
/  =  current,  E  =  impressed  e.m.f.,  consisting  of  two  com- 
ponents, one,  EI  =  E  cos  0,  in  phase  with  the  current,  the  other, 
1£2  =  E  sin  0,  in  quadrature  with  the  current,  the  power  in  the 
circuit  is  IEi  =  IE  cos  0,  and  the  e.m.f.  in  phase  with  the  current 
Ei  =  E  cos  0  is  a  power  e.m.f.,  the  e.m.f.  in  quadrature  with 
the  current  E2  =  E  sin  0  a  wattless  or  reactive  e.m.f. 


40       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

40.  Thus  we  have  to  distinguish  power  e.m.f.  and  wattless  or 
reactive  e.m.f.,  or  power  component  of  e.m.f.,  in  phase  with  the 
current  and  wattless  or  reactive  component  of  e.m.f.,  in  quadra- 
ture with  the  current. 

Any  e.m.f.  can  be  considered  as  consisting  of  two  components, 
one,  the  power  component,  e\,  in  phase  with  the  current,  and 
the  other,  the  reactive  component,  ez,  in  quadrature  with  the 
current.  The  sum  of  instantaneous  values  of  the  two  compo- 
nents is  the  total  e.m.f. 

e  =  ei  +  e* 
If  E}  EI,  Ez  are  the  respective  effective  values,  we  have 


E    =       Ei*  +  E22,  since 
EI  =  E  cos  &, 
E2  =  E  sin  6, 

where  0  =  phase  angle  between  current  and  e.m.f. 

Analogously,  a  current  I  due  to  an  impressed  e.m.f.  E  with 
a  time-phase  angle  0  can  be  considered  as  consisting  of  two 
component  currents, 

1 1  =  I  cos  8,  the  active  or  power  component  of  the  current,  and 
J2  =  /  sin  0,  the  wattless  or  reactive  component  of  the  current. 

The  sum  of  instantaneous  values  of  the  power  and  reactive 
components  of  the  current  equals  the  instantaneous  value  of  the 
total  current, 

ii  +  iz  =  i, 

while  their  effective  values  have  the  relation 

i  =  V77+772. 

Thus  an  alternating  current  can  be  resolved  in  two  com- 
ponents, the  power  component,  in  phase  with  the  e.m.f.,  and  the 
wattless  or  reactive  component,  in  quadrature  with  the  e.m.f. 

An  alternating  e.m.f.  can  be  resolved  in  two  components: 
the  power  component,  in  phase  with  the  current,  and  the  watt- 
less or  reactive  component,  in  quadrature  with  the  current. 

The  power  in  the  circuit  is  the  current  times  the  e.m.f.  times 
the  cosine  of  the  time-phase  angle,  or  is  the  power  component 
of  the  current  times  the  total  e.m.f.,  or  the  power  component  of 
the  e.m.f.  times  the  total  current. 


VECTOR  DIAGRAMS  41 

EXAMPLES 

41.  (1)  What  is  the  power  received  over  the  transmission  line 
in  Section  7,  Example  2,  the  power  lost  in  the  line,  the  power 
put  into  the  line,  and  the  efficiency  of  transmission  with  non- 
inductive  load,  with  45-time-degree  lagging  load  and  45-degree 
leading  load? 

The  power  received  per  line  with  non-inductive  load  is  P  =  El 
=  3170  X  44  =  139  kw. 

With  a  load  of  45  degrees  phase  displacement,  P  =  El  cos 
45°  =  98  kw. 

The  power  lost  per  line  PI  =  PR  =  442  X  7.6  =  14.7  kw. 

Thus  the  input  into  the  line  P0  =  P  +  PI  =  151.7  kw.  at 
non-inductive  load, 

and       =  111.7  kw.  at  load  of  45  degrees  phase  displacement. 
The  efficiency  with  non-inductive  load  is 

P  14  7 

Po  =  l  -  15T7  =    )0-3  p 

and  with  a  load  of  45  degrees  phase  displacement  is 

P  14.7 

^-  =  1  —  111  -,  =  86.8  per  cent. 

L  Q  111./ 

The  total  output  is  3  P  =  411  kw.  and  291  kw.,  respectively. 
The  total  input  3  P0  =  451.1  kw.  and  335.1  kw.,  respectively. 

9.  VECTOR  DIAGRAMS 

42.  The  best  way  of  graphically  representing  alternating-cur- 
rent phenomena  is  by  a  vector  diagram.     The  most  frequently 
used  vector  diagram  is  the  crank  diagram.     In  this,  sine  waves 
of  alternating  currents,  voltages,  etc.,  are  represented  as  projec- 
tions of  a  revolving  vector  on  the  horizontal.     That  is,  a  vector 
equal  in  length  to  the  maximum  value  of  the  alternating  wave  is 
assumed  to  revolve  at  uniform  speed  so  as  to  make  one  complete 
revolution  per  period,  and  the  projections  of  this  revolving  vec- 
tor upon  the  horizontal  then  represent  the  instantaneous  values 
of  the  wave. 

Let,  for  instance,  01  represent  in  length  the  maximum  value 
of  current  i  =  I  cos  (6  —  00).  Assume  then  a  vector,  07,  to 
revolve,  left-handed  or  in  positive  direction,  so  that  it  makes  a 


42       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

complete  revolution  during  each  cycle  or  period.  If  then  at  a 
certain  moment  of  time  this  vector  stands  in  position  OIi  (Fig. 
14),  the  projection,  OA^'  of  Oh  on  OA  represents  the  instan- 
taneous value  of  the  current  at  this  moment.  At  a  later  moment 
07  has  moved  farther,  to  0/2,  and  the  projection,  OAZ,  of  072  on 
OA  is  the  instantaneous  value.  The  diagram  thus  shows  the 
instantaneous  condition  of  the  sine  waves.  Each  sine  wave 


FIG.  14. — Crank  diagram  showing 
instantaneous  values. 


FIG.  15. — Crank  diagram  of  an 
e.m.f.  and  current. 


reaches  the  maximum  at  the  moment  when  its  revolving  vector, 
01,  passes  the  horizontal,  and  reaches  zero  when  its  revolving 
vector  passes  the  vertical. 

If  Fig.  15  represents  the  crank  diagram  of  a  voltage  OE,  and 
a  current^O/,  and  if  angle  AOE^AOI,  this  means  that  the 
current  01  is  behind  the  voltage  OE,  passes  during  the  revolu- 
tion the  zero  line  or  line  of  maximum  intensity,  OA,  later  than 
the  voltage;  that  is,  the  current  lags  behind  the  voltage. 

In  the  vector  diagram,  the  first  quantity  therefore  can  be  put 
in  any  position.  For_instance,  the  current  01,  in  Fig.  15,  could 
be  drawn  in  position  01,  Fig.  16.  The  voltage  then  being  ahead 


VECTOR  DIAGRAMS 


43 


of  the  current  by  angle  EOI  =  0  would  come  into  the  position  OE, 
Fig.  16. 

This  vector  diagram  then  shows  graphically,  by  the  projections 
of  the  vectors  on  the  horizontal,  the  instantaneous  values  of  the 
alternating  waves  at  one  moment  of  time.  At  any  other  moment 


FIG.  16. — Crank  diagram. 

of  time,  the  instantaneous  values  would  be  the  projections  of  the 
vectors  on  another  radius,  corresponding  to  the  other  time.  The 
angles  between  the  vector  representation  are  the  phase  differ- 
ences between  the  vectors,  and  the  angles  each  vector  makes  with 
the  horizontal  may  be  called  its  phase.  The  horizontal  then 


FIG.  17. — Vector  diagram  of  two  e.m.f.'s  acting  in  the  same  circuit. 

would  be  of  phase  zero.     The  phase  of  the  first  vector  may  be 
chosen  at  random;  all  other  phases  are  determined  thereby. 

In  this  representation,  the  phase  of  an  alternating  wave  is 
given  by  the  time  when  its  maximum  value  passes  the  horizontal. 


44       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Two  voltages,  e\  and  e2,  acting  in  the  same  circuit,  give  a 
resultant  voltage  e  equal  to  the  sum  of  their  instantaneous  values. 
Graphically,  voltages  ei  and  e%  are  represented  in  intensity  and 
in  phase  by  two  revolving  vectors,  OEi  and  OEZ,  Fig.  17.  The 
instantaneous  values  are  the  projections  Oei,  Oe2  of  OEi  and  OEZ 
upon  the  horizontal. 

Since  the  sum  of  the  projections  of  the  sides  of  a  parallelogram 
is  equal  to  thejDrojection  of  the  diagonal,jthe  sum  of  the  projec- 
tions Oei  and  Oez  equals  the  projection  Oe  of  OE,  the  diagonal 
of  the  parallelogram  with  OEi  and  OEZ  as  sides,  and  OE  is  thus 
the  resultant  e.m.f . ;  that  is,  graphically  alternating  sine  waves  of 
voltage,  current,  etc.,  are  combined  and  resolved  by  the  parallelo- 
gram or  polygon  of  sine  waves. 


FIG.  18. — Vector  diagram. 

43.  The  sine  wave  of  alternating  current  i  =  I0  sin  0  is  repre- 
sented by  a  vector  equal  in  length,  01 0,  to  the  maximum  value  70 
of  the  wave,  and  located  so  that  at  time  zero  0=0,  its  projec- 
tion on  the  horizontal,  is  zero,  and  at  times  0  >  0,  but  <  TT,  the 
projection  is  positive.  Thus  this  vector  0/0  is  the  negative 
vertical,  as  shown  in  Fig.  18. 

The  voltage  consumed  by  inductance,  ez  =  x!0  cos  0,  is  repre- 
sented by  a  vector  OEZ  equal  in  length  to  x!Q,  and  located  so 
that  at  0  =  0,  its  projection  on  the  horizontal  is  a  maximum. 
That  is,  it  is  the  zero  vector  OE2  in  Fig.  18. 

Analogously,  the  counter  e.m.f.  of  self-inductance  E'2  is 
represented  by  vector  OE'Z  on  the  negative  horizontal  of  Fig. 
18;  the  voltage  consumed  by  the  resistance  r,  e\  —  e!Q  sin  0,  is 
represented  by  vector  OEi  equal  to  r/0,  and  located  on  the  nega- 


VECTOR  DIAGRAMS  45 

tive  vertical,   and  the   counter  e.m.f.  of  resistance  by  vector 
OE'i  on  the  positive  vertical. 
The  counter  e.m.f.  of  impedance: 


—  (r/o  sin  0  +  x!Q  cos  0) 

-   ?Jn  sin   (ft  -\-  fi»} 


sin  (6  +  00) 


then  is  represented  graphically  as  the  resultant,  by  the  parallelo- 
gram of  sine  waves  of  OE\  and  OE'2}  that  is,  by  a  vector  OE', 
equal  in  length  to  z!0,  and  of  phase  90  +  00. 

The  voltage  consumed  by  impedance,  or  the  impressed  voltage, 
is  represented  by  the  vector  OE,  equal  and  opposite  in  direction 
to  the  vector  OE' .  This  vector  is  the  resultant  of  OEi  and  OE2 
and  has  the  phase  00  —  90,  or  —  (90  —  00),  as  shown  in  Fig.  18. 

An  alternating  wave  is  thus  determined  by  the  length  and  direc- 
tion of  its  vector.  The  length  is  the  maximum  value,  intensity  or 
amplitude  of  the  wave;  the  direction  is  the  phase  of  its  maximum 
value,  usually  called  the  phase  of  the  wave. 

44.  As  phase  of  the  first  quantity  considered,  as  in  the  above 
instance  the  current,  any  direction  can  be  chosen.  The  further 
quantities  are  determined  thereby  in  direction  or  phase. 

The  zero  vector  OA  is  generally  chosen  for  the  most  frequently 
used  quantity  or  reference  quantity,  as  for  the  current,  if  a  num- 
ber of  e.m.fs.  are  considered  in  a  circuit  of  the  same  current,  or 
for  the  e.m.f.,  if  a  number  of  currents  are  produced  by  the  same 
e.m.f.,  or  for  the  generated  e.m.f.  in  apparatus  such  as  transform- 
ers and  induction  motors,  synchronous  apparatus,  etc. 

With  the  current  as  zero  vector,  all  horizontal  components  of 
e.m.f.  are  power  components,  all  vertical  components  are  reac- 
tive components. 

With  the  e.m.f.  as  zero  vector,  all  horizontal  components  of 
current  are  power  components,  all  vertical  components  of  current 
are  reactive  components. 

By  measurement  from  the  vector  diagram  numerical  values 
can  hardly  ever  be  derived  with  sufficient  accuracy,  since  the 
magnitudes  of  the  different  quantities  used  in  the  same  diagram 
are  usually  by  far  too  different,  and  the  vector  diagram  is  there- 
fore useful  only  as  basis  for  trigonometrical  or  other  calculation, 
and  to  give  an  insight  into  the  mutual  relation  of  the  different 
quantities,  and  even  then  great  care  has  to  be  taken  to  distinguish 
between  the  two  equal  but  opposite  vectors,  counter  e.m.f.  and 
e.m.f.  consumed  by  the  counter  e.m.f.,  as  explained  before. 


46       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

EXAMPLES 

45.  In  a  three-phase  long-distance  transmission  line,  the  vol- 
tage between  lines  at  the  receiving  end  shall  be  5000  at  no  load, 
5500  at  full  load  of  44  amp.  power  component,  and  propor- 
tional at  intermediary  values  of  the  power  component  of  the 
current;  that  is,  the  voltage  at  the  receiving  end  shall  increase 
proportional  to  the  load.  At  three-quarters  load  the  current 
shall  be  in  phase  with  the  e.m.f.  at  the  receiving  end.  The 
generator  excitation,  however,  and  thus  the  (nominal)  generated 


FIG.  19. — Vector  diagram  of  e.m.f.  and  current  in  transmission  line.     Cur- 
rent leading. 


e.m.f.  of  the  generator  shall  be  maintained  constant  at  all  loads, 
and  the  voltage  regulation  effected  by  producing  lagging  or 
leading  currents  with  a  synchronous  motor  in  the  receiving  cir- 
cuit. The  line  has  a  resistance  rx  =  7.6  ohms  and  a  reactance 
Xi  =  4.35  ohms  per  wire,  the  generator  is  star  connected,  the 
resistance  per  circuit  being  r2  =  0.71,  and  the  (synchronous) 
reactance  is  x2  =  25  ohms.  ^  What  must  be  the  wattless  or  re- 
active component  of  the  current,  and  therefore  the  total  current 
and  its  phase  relation  at  no  load,  one-quarter  load,  one-half  load, 
three-quarters  load,  and  full  load,  and  what  will  be  the  terminal 
voltage  of  the  generator  under  these  conditions? 

The  total  resistance  of  the  line  and  generator  is  r  =  TI  +  r2 
=  8.31  ohms;  the  total  reactance,  x  =  Xi  +  #2  =  29.35  ohms. 

Let,  in  the  polar  diagram,  Fig.  19  or  20,  OE  =  E  represent 
the  voltage  at  the  receiving  end  of  the  line,  OIi  =  I\  the  power 
component  of  the  current  corresponding  to  the  load,  in  phase 
with  OE,  and  0/2  =  Iz  the  reactive  component  of  the  current 
in  quadrature  with  OE,  shown  leading  in  Fig.  19,  lagging  in  Fig. 
20. 

We  then  have  total  current  /  =  01. 


VECTOR  DIAGRAMS 


47 


Thus  the  e.m.f.  consumed  by  resist ance_,j9£'i  =  rl,  is  in  phase 
with  7,the  e.m.f.  consumed  by  reactance,  OEz  =  xl,  is  90  degrees 
ahead  of  /,  and  their  resultant  is  OE3,  the  e.m.f.  consumed  by 
impedance. 

OW3  combined  with  0#,  the  receiver  voltage,  gives  the  genera- 
tor voltage  OE0. 


FIG.  20. — Vector  diagram  of  e.m.f.  and  current  in  transmission  line.     Cur- 
rent lagging. 

Resolving  all  e.m.fs.  and  currents  into  components  in  phase 
and  in  quadrature  with  the  received  voltage  E,  we  have 


Current 

E.m.f.  at  receiving  end  of  line,  E  = 
E.m.f.  consumed  by  resistance,  EI  = 
E.m.f.  consumed  by  reactance,  E2  = 

Thus  total  e.m.f.  or  generator  voltage, 
E0  =  E  +  E!  +  E2  =  E  + 

Herein  the  reactive  lagging  component  of  current  is  assumed 
as  positive,  the  leading  as  negative. 

The  generator  e.m.f.  thus  consists  of  two  components,  which 
give  the  resultant  value 


Phase 
component 

Quadrature 
component 

/I 

~/2 

E 

0 

r/i 

-r/2 

xlz 

+   Z/1 

+  xI 


xll  -  r/ 


E,  =  V(E  +  rh  +  xI,Y  +  (xh  -  r/2)2; 
substituting  numerical  values,  this  becomes 


+  8.31 
At  three-quarters  load, 
5375 


+  29.35  72)2  +  (29.35  A  -  8.31  72)2. 


E  = 


3090  volts  per  circuit, 


48       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

/i  =  33,  72  =  0,  thus 

Eo  =  \/(3090  +  8.31  X  33)2  +  (29.35  X  33)2  =  3520  volts 
per  line  or  3520  X  \/3  =  6100  volts  between  lines 
as  (nominal)  generated  e.m.f.  of  generator. 

Substituting  these  values,  we  have 

3520  =  \/(E  +  8.31  7i  +  29.35  72)2  +  (8.31  12  -  29.35  /O2. 
The  voltage  between  the  lines  at  the  receiving  end  shall  be: 

No  U  M  H  Full 

load          load  load  load          load 

Voltage  between  lines,  5000     5125     5250     5375     5500 

Thus,  voltage  per  line  -5-  \/3,  #  =  2880    2950     3020     3090     3160 

The  power  components  of  current 

per  line,  II  =     0        11         22         33         44 
Herefrom  we  get  by  substituting  in  the  above  equation 

Reactive  component  of       £°d  lotd         lo^d         ^         ^d 

current,  72  =  -21.6     -16.2     -9.2         0          +9.7 

hence,  the  total  current, 


+  /22  =      21.6         19.6       23.9       33.0       45.05 
and  the  power  factor, 

^  =  cos  0  =  0         56.0       92.0     100.0       97.7 

the  lag  of  the  current, 

0  =      90°  61°          23°          0°       -11.5° 

the  generator  terminal  voltage  per  line  is 


E'  =  V(E  +  rj,  

=  V(E  +  7.6  A  +  4.35 72)2  +  (4.35 II  -  7.6  72)2 
thus: 

No  \i  M  H.          Full 

load          load          load          load  load 

Per  line,  E'_  =  2980     3106     3228     3344     3463 

Between  lines,  E'  V3  =  5200     5400     5600     5800     6000 

Therefore  at  constant  excitation  the  generator  voltage  rises  with 
the  load,  and  is  approximately  proportional  thereto. 

10.  HYSTERESIS  AND  EFFECTIVE  RESISTANCE 

46.  If  an  alternating  current  01  =  I,  in  Fig.  21,  exists  in  a 
circuit  of  reactance  x  =  2  irfL  and  of  negligible  resistance,  the 


HYSTERESIS  AND  EFFECTIVE  RESISTANCE      49 


magnetic  flux  produced  by  the  current,  0$  =  $,  is  in  phase  with 
the  current,  and  the  e.m.f.  generated  by  this  flux,  or  counter 
e.m.f.  of  self-inductance,  OE'"  =  E'"  =  xl,  lags  90  degrees  be- 
hind the  current.  The  e.m.f.  consumed  by  self-inductance  or 
impressed  e.m.f.  OE"  =  E"  =  xl  is  thus  90  degrees  ahead  of 
the  current. 

Inversely,  if  the  e.m.f.  OE"  =  E"  is  impressed  upon  a  circuit 
of  reactance  x  =  2  irfL  and  of  negligible  resistance,  the  current 

E" 

01  =  I  =  -  -  lags  90  degrees  behind  the  impressed  e.m.f. 
x 

This  current'  is  called  the  exciting  or  magnetizing  current  of 
the  magnetic  circuit,  and  is  wattless. 

'  If  the  magnetic  circuit  contains  iron  or  other  magnetic  mate- 
rial, energy  is  consumed  in  the  magnetic 
circuit    by   a  frictional    resistance  of   the 
material   against  a  change  of  magnetism, 
which  is  called  molecular  magnetic  friction. 

If  the  alternating  current  is  the  only  avail- 
able source  of  energy  in  the  magnetic  cir- 
cuit, the  expenditure  of  energy  by  molec- 
ular magnetic  friction  appears  as  a  lag  of 

the  magnetism  behind  the  m.m.f.  of  the     Q| r      >i 

current,  that  is,  as  magnetic  hysteresis,  and 
can  be  measured  thereby. 

Magnetic  hysteresis  is,  however,  a  dis- 
tinctly different  phenomenon  from  molec- 
ular magnetic  friction,  and  can  be  more 
or  less  eliminated,  as  for  instance  by  me- 
chanical vibration,  or  can  be  increased, 
without  changing  the  molecular  magnetic 
friction. 

47.  In  consequence  of  magnetic  hysteresis, 
if  an  alternating  e.m.f.  OE"  =  E"  is  im- 
pressed upon  a  circuit  of  negligible  resistance,  the  exciting 
current,  or  current  producing  the  magnetism,  in  this  circuit  is 
not  a  wattless  current,  or  current  of  90  degrees  lag,  as  in  Fig.  21, 
but  lags  less  than  90  degrees,  by  an  angle  90  —  a,  as  shown  by 
OI  =  I  in  Fig.  22. 

Since  the  magnetism  0$  =  $  is  in  quadrature  with  the  e.m.f. 
E"  due  to  it,  angle  a  is  the  phase  difference  between  the  magnet- 
ism and  the  m.m.f.,  or  the  lead  of  the  m.m.f.,  that  is,  the  exciting 

4 


FIG.  21.— Phase  re- 
lations of  magnetizing 
current,  flux  and  self- 
inductive  e.m.f. 


50       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

current,  before  the  magnetism.  It  is  called  the  angle  of  hysteretic 
lead. 

In  this  case  the  exciting  current  01  =  I  can  be  resolved  in  two 
components:  the  magnetizing  current  01 2  —  1 2,  in  phase  with 
the  magnetism  0$  =  $,  that  is,  in  quadrature  with  the  e.m.f. 
OE"  =  E"j  and  thus  wattless,  and  the  magnetic  power  component 
of  the  current  or  the  hysteresis  current  OIi  =  Ii,  in  phase jvvith  the 
e.m.f.  OE"  =  E",  or  in  quadrature  with  the  magnetism  0$  =  $. 

Magnetizing  current  and  hysteresis  current  are  the  two  com- 
ponents of  the  exciting  current. 


FIG.  22. — Angle  of  hysteretic 
lead. 


FIG.  23. — Effect  of  resistance 
on  phase  relation  of  impressed 
e.m.f.  in  a  hysteresisless  circuit. 


If  the  circuit  contains  besides  the  reactance  x  =  2  wfL,  a  re- 
sistance r,  the  e.m.f.  OE"  =  E"  in  the  preceding  Figs.  21  and  22 
is  not  the  impressed  e.m.f.,  but  the  e.m.f.  consumed  by  self- 
inductance  or  reactance,  and  has  to  be  combined,  Figs.  23  and 
24,  with  the  e.m.f.  consumed  by  the  resistance,  OE'  =  E'  =  Ir, 
to  get  the  impressed  e.m.f.  OE  =  E. 

Due  to  the  hysteretic  lead  a,  the  lag  of  the  current  is  less  in 
Figs.  22  and  24,  a  circuit  expending  energy  in  molecular  mag- 
netic friction,  than  in  Figs.  21  and  23,  a  hysteresisless  circuit. 

As  seen  in  Fig.  24,  in  a  circuit  whose  ohmic  resistance  is  not 
negligible,  the  hysteresis  current  and  the  magnetizing  current 
are  not  in  phase  and  in  quadrature  respectively  with  the  im- 
pressed e.m.f.,  but  with  the  counter  e.m.f.  of  inductance  or  e.m.f. 
consumed  by  inductance. 

Obviously  the  magnetizing  current  is  not  quite  wattless,  since 


HYSTERESIS  AND  EFFECTIVE  RESISTANCE     51 


energy  is  consumed  by  this  current  in  the  ohmic  resistance  of 
the  circuit. 

Resolving,  in  Fig.  25,  the  impressed  e.m.f.  OE  =  E  into  two 
components,  OEi  =  EI  in  phase,  and  OE2  =  E2  in  quadrature 
with  the  current  01  =  I,  the  power  component  of  the  e.m.f., 

EI,  is  greater  than  Er  =  Ir,  and  the  reactive  component 

E2  is  less  than  E" 


OE, 


Ix. 


FIG.  24. — Effect  of  resistance 
on  phase  relation  of  impressed 
e.m.f.  in  a  circuit  having  hys- 
teresis. 


FIG.  25. — Impressed  e.m.f.  resolved 
into  components  in  phase  and  in 
quadrature  with  the  exciting  current. 


The  value  r' 


ance,  and  the  value  x'  —  -r 


EI       power  e.m.f.  .  ...    . .          .  . 

-=-  =  *-  -  is  called  the  effective  resist- 

I        total  current 

E2       wattless  e.m.f. 
I 


is  called  the  ap- 


total  current 
parent  or  effective  reactance  of  the  circuit. 

48.  Due  to  the  loss  of  energy  by  hysteresis  (eddy  currents, 
etc.),  the  effective  resistance  differs  from,  and  is  greater  than, 
the  ohmic  resistance,  and  the  apparent  reactance  is  less  than  the 
true  or  inductive  reactance. 

The  loss  of  energy  by  molecular  magnetic  friction  per  cubic 
centimeter  and  cycle  of  magnetism  is  approximately 

W  =  r}B^, 

where  B    =  the  magnetic  flux  density,  in  lines  per  sq.  cm. 

W  =  energy,  in  absolute  units  or  ergs  per  cycle  (=  10~7 
watt-seconds  or  joules),  and  t\  is  called  the  coef- 
ficient of  hysteresis. 


52       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

In  soft  annealed  sheet  iron  or  sheet  steel  and  in  silicon  steel, 
rj  varies  from  0.60  X  10~3  to  2.5  X  10~3,  and  can  in  average,  for 
good  material,  be  assumed  as  1.5  X  10~3. 

The  loss  of  power  in  the  volume,  V,  at  flux  density  B  and 
frequency  /,  is  thus 

P  =  VfoB1'6  X  10"7,  in  watts, 

and,  if  /  =  the  exciting  current,  the  hysteretic  effective  resist- 
ance is 

P  B1'6 

r"  =J-*  =  VfrW-^' 

If  the  flux  density,  B,  is  proportional  to  the  current,  /,  sub- 
stituting for  B,  and  introducing  the  constant  k,  we  have 

rn          V 

~  'PA' 

that  is,  the  effective  hysteretic  resistance  is  inversely  propor- 
tional to  the  0.4  power  of  the  current,  and  directly  proportional 
to  the  frequency. 

49.  Besides  hysteresis,  eddy  or  Foucault  currents  contribute 
to  the  effective  resistance. 

Since  at  constant  frequency  the  Foucault  currents  are  pro- 
portional to  the  magnetism  producing  them,  and  thus  approxi- 
mately proportional  to  the  current,  the  loss  of  power  by  Foucault 
currents  is  proportional  to  the  square  of  the  current,  the  same  as 
the  ohmic  loss,  that  is,  the  effective  resistance  due  to  Foucault 
currents  is  approximately  constant  at  constant  frequency,  while 
that  of  hysteresis  decreases  slowly  with  the  current. 

Since  the  Foucault  currents  are  proportional  to  the  frequency, 
their  effective  resistance  varies  with  the  square  of  the  frequency, 
while  that  of  hysteresis  varies  only  proportionally  to  the 
frequency. 

The  total  effective  resistance  of  an  alternating-current  circuit 
increases  with  the  frequency,  but  is  approximately  constant, 
within  a  limited  range,  at  constant  frequency,  decreasing  some- 
what with  the  increase  of  magnetism. 

EXAMPLES 

50.  A  reactive  coil  shall  give  100  volts  e.m.f.  of  self-inductance 
at   10  amp.   and  60  cycles.     The  electric   circuit  consists  of 
200  turns  (No.  8  B.  &  S.)  (=  0.013  sq.  in.)  of  16  in.  mean  length 
of  turn.     The  magnetic  circuit  has  a  section  of  6  sq.  in.  and  a 


HYSTERESIS  AND  EFFECTIVE  RESISTANCE      53 

mean  length  of  18  in.  of  iron  of  hysteresis  coefficient  rj  =  2.5  X 
1CT3.  An  air  gap  is  interposed  in  the  magnetic  circuit,  of  a 
section  of  10  sq.  in.  (allowing  for  spread),  to  get  the  desired 
reactance. 

How  long  must  the  air  gap  be,  and  what  is  the  resistance,  the 
reactance,  the  effective  resistance,  the  effective  impedance,  and 
the  power-factor  of  the  reactive  coil? 

The  coil  contains  200  turns  each  16  in.  in  length  and  0.013 
sq.  in.  in  cross  section.  Taking  the  resistivity  of  copper  as  1.8  X 
10~6,  the  resistance  is 

1.8  X  10~6  X  200  X  16 

0.013  X  2.54"  im> 

where  2.54  is  the  factor  for  converting  inches  to  centimeters. 
(1  inch  =  2.54  cm.) 

Writing  E  =  100  volts  generated,  /  =  60  cycles  per  second, 
and  n  =  200  turns,  the  maximum  magnetic  flux  is  given  by 
E  =  4.44  fn$;  or,  100  =  4.44  X  0.6  X  200$,  and  3>  =  0.188 
megaline. 

This  gives  in  an  air  gap  of  10  sq.  in.  a  maximum  density 
B  =  18,800  lines  per  sq.  in.,  or  2920  lines  per  sq.  cm. 

Ten  amperes  in  200  turns  give  2000  ampere-turns  effective  or 
F  =  2830  ampere-turns  maximum. 

Neglecting  the  ampere-turns  required  by  the  iron  part  of  the 
magnetic  circuit  as  relatively  very  small,  2830  ampere-turns 
have  to  be  consumed  by  the  air  gap  of  density  B  =  2920. 

Since  D        4?rF 

loT 

the  length  of  the  air  gap  has  to  be 
47TX2830 


To*  =:  ToxWo  ==  L22  cm"  or  °'48  ln' 

With  a  cross  section  of  6  sq.  in.  and  a  mean  length  of  18  in., 
the  volume  of  the  iron  is  108  cu.  in.,  or  1770  cu.  cm. 


OOO 

The  density  in  the  iron,  BI  =  --  g  --  =  31,330  lines  per  sq. 

in.,  or  4850  lines  per  sq.  cm. 

With  an  hysteresis  coefficient  77  =  2.5  X  10~3,   and   density 
BI  =  4850,  the  loss  of  energy  per  cycle  per  cubic  centimeter  is 

W  =  i/fii1-6 

=  2.5  X  10-3  X  48501-6 
=  1980  ergs, 


54       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

and  the  hysteresis  loss  at/  =  60  cycles  and  the  volume  V  =  1770 
is  thus 

P  =  60  X  1770  X  1980  ergs  per  sec. 
=  21.0  watts, 

which  at  10  amp.  represent  an  effective  hysteretic  resistance, 

21.0 
r2  =  -y~j-  —  0.21  ohm. 

Hence  the  total  effective  resistance  of  the  reactive  coil  is 

r  =  n  +  r2  =  0.175  +  0.21  =  0.385  ohm 
the  effective  reactance  is 

777 

x  =  ~j  =  10  ohms; 

the  impedance  is 

z  =  10.01  ohms; 
the  power-factor  is 

T 

cos  0  —  -  =  3.8  per  cent.; 

z 

the  total  apparent  power  of  the  reactive  coil  is 

I2z  =  1001  volt-amperes, 
and  the  loss  of  power, 

Pr  =  38  watts. 

11.  CAPACITY  AND  CONDENSERS 

51.  The  charge  of  an  electric  condenser  is  proportional  to  the 
impressed  voltage,  that  is,  potential  difference  at  its  terminals, 
and  to  its  capacity. 

A  condenser  is  said  to  have  unit  capacity  if  unit  current  exist- 
ing for  one  second  produces  unit  difference  of  potential  at  its 
terminals. 

The  practical  unit  of  capacity  is  that  of  a  condenser  in  which 
1  amp.  during  one  second  produces  1  volt  difference  of  potential. 

The  practical  unit  of  capacity  equals  10~9  absolute  units.  It 
is  called  a  farad. 

One  farad  is  an  extremely  large  capacity,  and  therefore  one 
millionth  of  one  farad,  called  microfarad,  mf.,  is  commonly  used. 

If  an  alternating  e.m.f.  is  impressed  upon  a  condenser,  the 
charge  of  the  condenser  varies  proportionally  to  the  e.m.f.,  and 


CAPACITY  AND  CONDENSERS  55 

thus  there  is  current  to  the  condenser  during  rising  and  from  the 
condenser  during  decreasing  e.m.f.,  as  shown  in  Fig.  26. 

That  is,  the  current  consumed  by  the  condenser  leads  the 
impressed  e.m.f.  by  90  time  degrees,  or  a  quarter  of  a  period. 

Denoting  /  as  frequency  and  E  as  effective  alternating  e.m.f. 
impressed  upon  a  condenser  of  C'mf.  capacity,  the  condenser  is 
charged  and  discharged  twice  during  each  cycle,  and  the  time 

of  one  complete  charge  or  discharge  is  therefore  j^- 

Since  E  \/2  is  the  maximum  voltage  impressed  upon  the  con- 
denser, an  average  of  CE  \/2  10~6  amp.  would  have  to  exist 
during  one  second  to  charge  the  condenser  to  this  voltage,  and 


FIG.  26. — Charging  current  of  a  condenser  across  which  an  alternating  e.m.f. 

is  impressed. 

to  charge  it  in  j^  seconds  an  average  current  of  4  fCE  \/2  10~6 

amp.  is  required. 

effective  current         TT 
Since 


average  current  2\/2' 
the  effective  current  is  I  =  2-irfCE  10~6;  that  is,  at  an  impressed 
e.m.f.  of  E  effective  volts  and  frequency  /,  a  condenser  of  C  mf. 
capacity  consumes  a  current  of 

1  =  2  irfCE  10~6  amp.  effective, 

which   current  leads  the  terminal  voltage  by  90  degrees  or  a 
quarter  period. 

Transposing,  the  e.m.f.  of  the  condenser  is 

106/ 


106 
The  value  z0  =       fn  is  called  the  condensive  reactance  of  the 

^  7T/C 

condenser. 


56       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Due  to  the  energy  loss  in  the  condenser  by  dielectric  hysteresis, 
the  current  leads  the  e.m.f.  by  somewhat  less  than  90  time  de- 
grees, and  can  be  resolved  into  a  wattless  charging  current  and  a 
dielectric  hysteresis  current,  which  latter,  however,  is  generally 
so  small  as  to  be  negligible,  though  in  underground  cables  of 
poor  quality,  it  may  reach  as  high  as  50  per  cent,  or  more  of 
the  charging  or  wattless  current  of  the  condenser. 

52.  The  capacity  of  one  wire  of  a  transmission  line  is 

i.nxio-6x/  . 

C  =  -  —  ~-i  -  ,  in  mf., 


where  Id  =  diameter  of  wire,  cm.;  18  —  distance  of  wire  from 
return  wire,  cm.;  I  =  length  of  wire,  cm.,  and  1.11  X  10~6  = 
reduction  coefficient  from  electrostatic  units  to  mf  . 

The  logarithm  is  the  natural  logarithm;  thus  in  common  loga- 
rithms, since  loge  a  =  2.303  logio  a,  the  capacity  is 

0.24  X  10~6  X  I 


i  ^  t>s 

logio  -7- 

I'd 


.        , 
,  in  mf  . 


The  derivation  of  this  equation  must  be  omitted  here. 
The  charging  current  of  a  line  wire  is  thus 

1  =  2  7T/CE  10~6, 

where  /  =  the  frequency,  in  cycles  per  second,  E  =  the  difference 
of  potential,  effective,  between  the  line  and  the  neutral  (E  — 
y^  line  voltage  in  a  single-phase,  or  four-wire  quarter-phase  sys- 

tem, —  -i=.  line  voltage,  or  Y  voltage,  in  a  three-phase  system). 
V  3 

EXAMPLES 

53.  In  the  transmission  line  discussed  in  the  examples  in 
37,  38,  41  and  45,  what  is  the  charging  current  of  the  line  at  6000 
volts  between  lines,  at  33.3  cycles?  How  many  volt-amperes 
does  it  represent,  and  what  percentage  of  the  full-load  current  of 
44  amp.  is  it? 

The  length  of  the  line  is,  per  wire,  I    =  2.23  X  106  cm. 
The  distance  between  wires,  ls  =  45  cm. 

The  diameter  of  transmission  wire,  Id  =  0.82  cm. 
Thus  the  capacity,  per  wire,  is 


C  =     -  .  0.26  mf. 

1  <£  ts 

loglo  -• 


IMPEDANCE  OF  TRANSMISSION  LINES          57 

The  frequency  is  /  =  33.3, 

The  voltage  between  lines,  6000. 

Thus  per  line,  or  between  line  and  neutral  point, 


E  =  =  3460  volts; 


hence,  the  charging  current  per  line  is 

Jo  =  2  irfCE  10  ~6 

=  0.19  amp., 
or  0.43  per  cent,  of  full-load  current; 

that  is,  negligible  in  its  influence  on  the  transmission  voltage. 
The  volt-ampere  input  of  the  transmission  is, 

3  IQE  =  2000 

=  2.0  kv-amp. 

12.  IMPEDANCE  OF  TRANSMISSION  LINES 

54.  Let  r  =  resistance;  x  =  2  irfL  =  the  reactance  of  a  trans- 
mission line;  E0  =  the  alternating  e.m.f.  impressed  upon  the  line; 
I  =  the  line  current;  E  =  the  e.m.f. 
at  receiving  end  of  the  line,  and  6  = 
the  angle  of  lag  of  current  7  behind 
e.m.f.  E. 

B  <  0  thus  denotes  leading,  0  >  0 
lagging  current,  and  6  =  0  a  non-in- 
ductive receiver  circuit. 

The  capacity  of  the  transmission   0 

line  shall  be  considered  as  negligible.       FIG.  27.—  Vector   diagram 
,1         i  f  ,  v  i.     of    current   and  e.m.fs.   in  a 

_Assummg  the  phase  of  the  current    transmission    line    assuming 
QI  =  /  as  zero  in  the  polar  diagram,    zero  capacity. 
Fig.  27,  the  e.m.f.  E  is  represented  by 

vector  OE,  ahead  of  07  by  angle  0.  The  e.m.f.  consumed  by  re- 
sistance r  is  OEi  =  Ei  =  Ir  in  phase  with  the  current,  and  the 
e.m.f.  consumed  by  reactance  x  is  OE%  =  Ez  =  Ix,  90  time  de- 
grees ahead  of  the  current;  thus  the  total  e.m.f.  consumed  by  the 
line,  or  e.m.f.  consumed  by  impedance,  is  the  resultant  OES  of 


and  O#2,  jind  is  E3  =  Iz. 
Combining  OEz  and  OE  gives  OEQ,  the  e.m.f.  impressed  upon 
the  line. 


58       ELEMENTS  OF  ELECTRICAL  ENGINEERING 


Denoting  tan  0i  =  -  the  time  angle  of  lag  of  the  line  impe- 
dance, it  is,  trigonometrically, 


Since 


OE02  =  OE2  +  EEQ2  -  2  OE  X  EEQ  cos 

~EEo  =  OE*  =  Iz, 
OEEQ  =  180  -  0i  +  6, 


FIG.  28. — Locus  of  the  generator  and  receiver  e.m.fs.  in  a  transmission  line 
with  varying  load  phase  angle. 


E02  =  E2  +  I2z2  +  2  EIz  cos  (0!  -  6) 
=  (E  +  Iz)2  -  4  #/z  sin2  ^-^, 


we  have 

and 

E0  =  \I(E  -f-  Iz)2  —  4  EIz  sin2  -^—= — , 

and  the  drop  of  voltage  in  the  line, 

EQ  -  E  =  \  (E  +  Iz}2  -  4  EIz  sin2  -^ E. 


IMPEDANCE  OF  TRANSMISSION  LINES 


59 


65.  That  is,  the  voltage  EQ  required  at  the  sending  end  of  a 
line  of  resistance  r  and  reactance  x,  delivering  current  /  at  vol- 
tage E}  and  the  voltage  drop  in  the  line,  do  not  depend  upon 
current  and  line  constants  only,  but  depend  also  upon  the  angle 
of  phase  displacement  of  the  current  delivered  over  the  line. 

If  0  =  o,  that  is,  non-inductive  receiving  circuit, 


FIG.  29. — Locus  of  the  generator  and  receiver  e.m.fs.  in  a  transmission  line 
with  varying  load  phase  angle. 


E0  = 


-  4  EIz  sin21; 


that  is,  less  than  E  +  Iz,  and  thus  the  line  drop  is  less  than  Iz. 

If  0  —  6  1,  EQ  is  a  maximum,  =  E  +  Iz,  and  the  line  drop  is  the 
impedance  voltage. 

With  decreasing  0,  E0  decreases,  and  becomes  =  E;  that  is, 
no  drop  of  voltage  takes  place  in  the  line  at  a  certain  negative 


60      ELEMENTS  OF  ELECTRICAL  ENGINEERING 

value  of  0  which  depends  not  only  on  z  and  0i  but  on  E  and  7. 
Beyond  this  value  of  6,  EQ  becomes  smaller  than  E;  that  is,  a 
rise  of  voltage  takes  place  in  the  line,  due  to  its  reactance.  This 
can  be  seen  best  graphically. 

Choosing  the  current  vector  01  as  the  horizontal  axis,  for 
the  same  e.m.f.  E  received,  but  different  phase  angles  6,  all 
vectors  OE  lie  on  a  circle  e  with  0  as  center.  Fig.  28.  Vector 
OEz  is  constant  for  a  given  line  and  given  current  7. 

Since  E3EQ  =  OE  =  constant,  E0  lies  on  a  circle  eQ  with  Es  as 
center  and  OE  =  E  as  radius. 

To  construct  the  diagram  for  angle  0,  OE  is  drawn  at  the  angle 
0  with  07,  and  EEQ  parallel  to  ~OE*. 

The  distance  E±EQ  between  the  two  circles  on  vector  OEo  is 
the  drop  of  voltage  (or  rise  of  voltage)  in  the  line. 

As  seen  in  Fig.  29,  EQ  is  maximum  in  the  direction  OE%  as  OE'0, 
that  is,  for  0  =  00,  and  is  less  for  greater  as  well,  OE"  'o,  as  smaller 
angles  6.  It  is  =  E  in  the  direction  072"'o,  in  which  case  0  <  0, 
and  minimum  in  the  direction 


The  values  of  E  corresponding  to  the  generator  voltages  E'Q, 
E"0,  E'"Q,  #IV0  are  shown  by  the  points  E'  E"  Ef"  E™  respectively. 
The  voltages  E"Q  and  Elv$  correspond  to  a  wattless  receiver  cir- 
cuit E"  and  E™.  For  non-inductive  receiver  circuit  W  the 
generator  voltage  is  OEvo. 

56.  That  is,  in  an  inductive  transmission  line  the  drop  of 
voltage  is  maximum  and  equal  to  Iz  if  the  phase  angle  0  of  the 
receiving  circuit  equals  the  phase  angle  00  of  the  line.  The  drop 
of  voltage  in  the  line  decreases  with  increasing  difference  be- 
tween the  phase  angles  of  line  and  receiving  circuit.  It  becomes 
zero  if  the  phase  angle  of  the  receiving  circuit  reaches  a  certain 
negative  value  (leading  current).  In  this  case  no  drop  of  vol- 
tage takes  place  in  the  line.  If  the  current  in  the  receiving  cir- 
cuit leads  more  than  this  value  a  rise  of  voltage  takes  place  in 
the  line.  Thus  by  varying  phase  angle  9  of  the  receiving  circuit 
the  drop  of  voltage  in  a  transmission  line  with  current  7  can  be 
made  anything  between  Iz  and  a  certain  negative  value.  Or 
inversely  the  same  drop  of  voltage  can  be  produced  for  different 
values  of  the  current  7  by  varying  the  phase  angle.  / 

Thus,  if  means  are  provided  to  vary  the  phase  angle  of  the 
receiving  circuit,  by  producing  lagging  and  leading  currents  at 
will  (as  can  be  done  by  synchronous  motors  or  converters)  ,  the 
voltage  at  the  receiving  circuit  can  be  maintained  constant 


IMPEDANCE  OF  TRANSMISSION  LINES 


61 


within  a  certain  range  irrespective  of  the  load  and  generator 
voltage. 

In  Fig.  30  let  OE  =  E,  the  receiving  voltage;  /,  the  power 
component  of  the  line  current;  thus  OES  =  Es  =  Iz,  the  e.m.f. 
consumed  by  the  power  component  of  the  current  in  the  impe- 
dance. This  e.m.f.  consists  of  the  e.m.f  consumed  by  resistance 
^Ei  and  the  e.m.f.  consumed  by  reactance  OEz- 


FIG.  30. — Regulation  diagram  for  transmission  line. 

Reactive  components  of  the  current  are  represented  in  the 
diagram  in  the  direction  OA  when  lagging  and  OB  when  leading. 
The  e.m.f.  consumed  by  these  reactive  components  of  the  current 
in  the  impedance  is  thus  in  the  direction  e1 '3,  perpendicular  to  OEs- 
Combining  OEz  and  OE  gives  the  e.m.f.  OE*  which  would  be 
required  for  non-inductive  load.  If  EQ  is  the  generator  voltage, 
EQ  lies  on  a  circle  eQ  with  O^o  as  radius.  Thus  drawing  E^E0  par- 
allel to  e'z  gives  OEQ,  the  generator  voltage;  OE'S  =  EJZo,  the 


62       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

e.m.f.  consumed  in  the  impedance  by  the  reactive  component  of 
the  current;  and  as  proportional  thereto,  OI'  =  I',  the  reactive 
current  required  to  give  at  generator  voltage  E0  and  power  cur- 
rent 7  the  receiver  voltage  E.  This  reactive  current  7'  lags  be- 
hind E'z  by  less  than  90  and  more  than  zero  degrees. 

57.  In  calculating  numerical  values,  we  can  pro'ceed  either 
trigonometrically  as  in  the  preceding,  or  algebraically  by  resolv- 
ing all  sine  waves  into  two  rectangular  components;  for  instance, 
a  horizontal  and  a  vertical  component,  in  the  same  way  as  in 
mechanics  when  combining  forces. 

Let  the  horizontal  components  be  counted  positive  toward 
the  right,  negative  toward  the  left,  and  the  vertical  components 
positive  upward,  negative  downward. 

Assuming  the  receiving  voltage  as  zero  line  or  positive  hori- 
zontal line,  the  power  current  7  is  the  horizontal,  the  wattless 
current  I'  the  vertical  component  of  the  current.  The  e.m.f.  con- 
sumed in  resistance  by  the  power  current  7  is  a  horizontal  com- 
ponent, and  that  consumed  in  resistance  by  the  reactive  current 
/'  a  vertical  component,  and  the  inverse  is  true  of  the  e.m.f. 
consumed  in  reactance. 

We  have  thus,  as  seen  from  Fig.  30: 

Horizontal  Vertical 

component  component 

Receiver  voltage,  E,  +  E  0 

Power  current,  /,  +  7  0 

Reactive  current,  7',  0  HF  7' 

E.m.f.  consumed  in  resistance  r  by  the 

power  current,  Ir,  +  Ir  0 

E.m.f.  consumed  in  resistance  r  by  the 

reactive  current,  7'r,  0  +  7V 

E.m.f.  consumed  in  reactance  x  by  the 

power  current,  Ix,  0  +  Ix 

E.m.f.  consumed  in  reactance  x  by  the 

reactive  current,  I'x,  ±  I'x  0 

Thus,  total  e.m.f.  required,  or  impressed 

e.m.f.,  Eo,  E  +  Ir±  I'x     +  I'r  +  Ix; 

hence,  combined, 


Eo  =  ^/(E  +  Ir± 
or,  expanded, 


+  2  E  (Ir  ±  I'x)  +  (72  +  7'2)*2- 


IMPEDANCE  OF  TRANSMISSION  LINES          63 

From  this  equation  I'  can  be  calculated ;  that  is,  the  reactive 
current  found  which  is  required  to  give  E0  and  E  at  energy 
current  7. 

The  lag  of  the  total  current  in  the  receiver  circuit  behind  the 
receiver  voltage  is 

tan  0  =  j. 

The  lead  of  the  generator  voltage  ahead  of  the  receiver  voltage 
is 

vertical  component  of  EQ 
horizontal  component  of  EQ 


B  +  Ir±  7V 

and  the  lag  of  the  total  current  behind  the  generator  voltage  is 


As  seen,  by  resolving  into  rectangular  components  the  phase 
angles  are  directly  determined  from  these  components. 

The  resistance  voltage  is  the  same  component  as  the  current  to 
which  it  refers. 

The  reactance  voltage  is  a  component  90  time  degrees  ahead 
of  the  current. 

The  same  investigation  as  made  here  on  long-distance  trans- 
mission applies  also  to  distribution  lines,  reactive  coils,  trans- 
formers, or  any  other  apparatus  containing  resistance  and 
reactance  inserted  in  series  into  an  alternating-current  circuit. 

EXAMPLES 

58.  (1)  An  induction  motor  has  2000  volts  impressed  upon 
its  terminals;  the  current  and  the  power-factor,  that  is,  the 
cosine  of  the  angle  of  lag,  are  given  as  functions  of  the  output 
in  Fig.  31. 

The  induction  motor  is  supplied  over  a  line  of  resistance 
r  =  2.0  and  reactance  x  =  4.0. 

(a)  How  must  the  generator  voltage  eQ  be  varied  to  maintain 
constant  voltage  e  =  2000  at  the  motor  terminals,  and 

(b)  At   constant   generator  voltage  CQ  =  2300,  how  will  the 
voltage  at  the  motor  terminals  vary? 


64       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

We  have 

/  a*  —  ft 

—.  e  =  2000. 


63.4°. 


«=  -i  l(e  +  iz)2  —  4  ezz  sin2 


tan  0i  =  ~  =  2. 


cos  0  =  power-factor. 

Taking  i  from  Fig.  31  and  substituting,  gives  (a)  the  values 
of  e0  for  e  =  2000,  which  are  recorded  in  the  table,  and  plotted  in 
Fig.  31. 


JTPUT 


.10  20  30  40  50  60  70  80  90  100  110  .120  130  140  150  160  170  180  190  200 

FIG.  31. — Characteristics  of  induction  motor  and  variation  of  generator 
e.m.f.  necessary  to  maintain  constant  the  e.m.f.  impressed  upon  the  motor. 

(6)  At  the  terminal  voltage  of  the  motor  e  =  2000,  the  cur- 
rent is  i,  the  output  P,  the  generator  voltage  eQ.  Thus  at  gen- 
erator voltage  e'o  =  2300,  the  terminal  voltage  of  the  motor  is 


the  current  is 
and  the  power  is 


2300  . 


p, 


The  values  of  e',  i' ,  P'  are  recorded  in  the  second  part  of  the 
table  under  (6)  and  plotted  in  Fig.  32. 


IMPEDANCE  OF  TRANSMISSION  LINES 


65 


(a)  At  e  =  2000 

Thus, 

M 

(6)  Hence,  at  eo  =  2300 

Output, 
P  =  kw. 

Current, 
t 

Lag. 

Output, 

Current, 
»' 

Voltage, 

0 

12.0 

84.3° 

2048 

0. 

13.45 

2240 

5 

12.6 

72.6° 

2055 

6.25 

14.05 

2234 

10 

13.5 

62.6° 

2060 

12.4 

15.00 

2230 

15 

14.8 

54.6° 

2065 

18.6 

16.4 

2220 

20 

16.3 

47.9° 

2071 

24.4 

18.0 

2216 

30 

20.0 

37.8° 

2084 

36.3 

22.0 

2200 

40 

25.0 

32.8° 

2093 

48.0 

27.5 

2198 

50 

30.0 

29.0° 

2110 

59.5 

32.7 

2180 

69 

40.0 

26.3° 

2146 

78.5 

42.8 

2160 

102 

60.0 

24.5° 

2216 

110.2 

62.6 

2080 

132 

80.0 

25.8° 

2294 

131.0 

79.5 

1990 

160 

100.0 

28.4° 

2382 

149.0 

96.4 

1928 

180 

120.0 

31.8° 

2476 

156.5 

111.5 

1860 

200 

150.0 

36.9° 

2618 

155.0 

132.0 

1760 

7 


L 


IS    30    36    40    50    60    JO   80    90   100   110   120   130   !40 

FIG.  32. — Characteristics  of  induction  motor,  constant  generator  e.m.f. 
5 


66       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

69.  (2)  Over  a  line  of  resistance  r  =  2.0  and  reactance  x  = 
6.0  power  is  supplied  to  a  receiving  circuit  at  a  constant  voltage 
of  e  =  2000.  How  must  the  voltage  at  the  beginning  of  the 
line,  or  generator  voltage,  eQ,  be  varied  if  at  no  load  the  receiving 
circuit  consumes  a  reactive  current  of  i*  —  20  amp.,  this  reac- 
tive current  decreases  with  the  increase  of  load,  that  is,  of  power 
current  i\,  becomes  iz  =  0  at  i\  =  50  amp.,  and  then  as  leading 
current  increases  again  at  the  same  rate? 


REACT 
CURRE 


FIG.  33.  —  Variation  of  generator  e.m.f.  necessary  to  maintain  constant 
receiver  voltage  if  the  reactive  component  of  receiver  current  varies  propor- 
tional to  the  change  of  power  component  of  the  current. 


The  reactive  current, 


=  20  at 
=    0  at 


and  can  be  represented  by 


=    0, 
=  50, 


•*2=     l  -        20  =  20  -0.4  ti; 
the  general  equation  of  the  transmission  line  is 


£o  =  V  (e  H-  iir  -f  izx) 
=  V(2000  +  2n  + 
hence,  substituting  the  value  of  z*2, 


(2*2  - 


e0  =  V(2120  -  0.4  n)2  +  (40  -  6.8ti)a 


=  V4,496,000  +  46.4  if  -  2240  ^. 


ALTERNATING-CURRENT  TRANSFORMER         67 


Substituting  successive  numerical  values  for  ii  gives  the  values 
recorded  in  the  following  table  and  plotted  in  Fig.  33. 


ii 

eo 

0 

'2120 

20 

2114 

40 

2116 

60 

2126 

80 

2148 

100 

2176 

120 

2213 

140 

2256 

160 

2308 

180 

2365 

200 

2430 

13.  ALTERNATING-CURRENT  TRANSFORMER 

60.  The  alternating-current  transformer  consists  of  one  mag- 
netic circuit  interlinked  with  two  electric  circuits,  the  primary 
circuit  which  receives  energy,  and  the  secondary  circuit  which 
delivers  energy. 

Let  TI  =  resistance,    x\  =  2TrfSz  =  self-inductive    or  leakage 

reactance  of  secondary  circuit, 

r0  =  resistance,    XQ  =  2irfSi  =  self -inductive  or  leakage 
reactance  of  primary  circuit, 

where  S2  and  Si  refer  to  that  magnetic  flux  which  is  interlinked 
with  the  one  but  not  with  the  other  circuit. 


Let    a 


ratio  of  — — •  — -  turns  (ratio  of  transformation), 
primary 


An  alternating  e.m.f.  E0  impressed  upon  the  primary  electric 
circuit  causes  a  current,  which  produces  a  magnetic  flux  $  inter- 
linked with  primary  and  secondary  circuits.  This  flux  <l>  gener- 
ates e.m.fs.  EI  and  E{  in  secondary  and  in  primary  circuit, which 

Tjl 

are  to  each  other  as  the  ratio  of  turns,  thus  Ei  =  —  - 

Let  E  =  secondary  terminal  voltage,  I\  =  secondary  current, 
0i  =  lag  of  current  /i  behind  terminal  voltage  E  (where  B\  <  0 
denotes  leading  current). 

Denoting  then  in  Fig.  34  by  a  vector  OE  =  E  the  secondary 


68       ELEMENTS  OF  ELECTRICAL  ENGINEERING 


terminal  voltage,  01 1  =  l\  is  the  secondary  current  lagging  by 
the  angle  EOI  =  61. 

The  e.m.f.  consumed  by  the  secondary  resistance  7*1  is  OE'i  = 
E'i  =  Iiri  in  phase  with  /i. 

The  e.m.f.  consumed  by  the  secondary  reactance  Xi  is  OE"\  = 
E'\  =  I&i,  90  degrees  ahead  of  /i.  Thus  the  e.m.f.  con- 
sumed by  the  secondary  impedance  z\  =  Vn2  +  Xi2  is  the 
resultant  of  OE'i  and  OE"i,  or  OE"\  =  E"\  =JiZi. 

OE'"\  combined  with  the  terminal  voltage  OE  =  E  gives  the 
secondary  e.m.f.  OEi  =  E\. 

Proportional  thereto  by  the  ratio  of  turns  and  in  phase  there- 


FIG. 34.  —  Vector  diagram  of  e.m.fs.  and  currents  in  a  transformer. 
with  is  the  e.m.f.  generated  in  the  primary  OEi  =  Ef   where 


To  generate  e.m.f.  EI  and  Ei}  the  magnetic  flux  0$  =  <i>  is 
required,  90  time  degrees  ahead  of  OE\  and  OEi.  To  produce 
flux  $  the  m.m.f.  of  F  ampere-turns  is  required,  as  determined 
from  the  dimensions  of  the  magnetic  circuit,  and  thus  the 
primary  current  /oo,  represented  by  vector  O/oo,  leading  0$  by 
the  angle  a. 

Since  the  total  m.m.f.  of  the  transformer  is  given  by  the 
primary  exciting  current  70o,  there  must  be  a  component  of 
primary  current  /',  corresponding  to  the  secondary  current  /i, 
which  may  be  called  the  primary  load  current,  and  which  is 


ALTERNATING-CURRENT  TRANSFORMER 


69 


opposite  thereto  and  of  the  same  m.m.f.;  that  is,  of  the  intensity 
/'  =  a/i,  thus  represented  by  vector  01'  =  I'  =  a/i. 

O/oo,  the  primary  exciting  current,  and  the  primary  load 
current  O/',  or  component  of  primary  current  corresponding 
to  the  secondary  current,  combined,  give  the  total  primary 
current  O/o  =  /o- 

The  e.m.f.  consumed  by  resistance  in  the  primary  is  OE'Q  = 
E'Q  =  /Or0  in  phase  with  /0. 

The  e.m.f.  consumed  by  the  primary  reactance  is  OE"o  =  E"Q 
=  /0£o,  90  degrees  ahead  of  O/o. 

OE'Q  and  OE"o  combined  gives  OE'"Q,  the  e.m.f.  consumed  by 
the  primary  impedance. 


FIG.  35. — Vector  diagram  of  transformer  with  lagging  load  current. 

Equal  and  opposite  to  the  primary  counter-generated  e.m.f. 
OEi  is  the  component  of  primary  e.m.f.,  OEf,  consumed  thereby. 

OE'  combined  with  OE"'Q  gives  OEQ  =  EQj  the  primary  im- 
pressed e.m.f.,  and  angle  0o  =  -#o#/o,  the  phase  angle  of  the 
primary  circuit. 

Figs.  35,  36,  and  37  give  the  polar  diagrams  of  0i  =  45°  or 
lagging  current,  0i  =  zero  or  non-inductive  circuit,  and  6  = 
—  45°  or  leading  current. 

61.  As  seen,  the  primary  impressed  e.m.f.  E0  required  to  pro- 
duce the  same  secondary  terminal  voltage  E  at  the  same  current 
1 1  is  larger  with  lagging  or  inductive  and  smaller  with  leading 


70       ELEMENTS  OF  ELECTRICAL  ENGINEERING 


current  than  on  a  non-inductive  secondary  circuit;  or,  inversely, 
at  the  same  secondary  current  I\  the  secondary  terminal  voltage 
E  with  lagging  current  is  less  and  with  leading  current  more 
than  with  non-inductive  secondary  circuit,  at  the  same  primary 
impressed  e.m.f.  EQ. 

The  calculation  of  numerical  values  is  not  practicable  by 
measurement  from  the  diagram,  since  the  magnitudes  of  the 
different  quantities  are  too  different,  E'i:E"iiEi:Eo  being 
frequently  in  the  proportion  1  :  10  :  100  :2000. 

Trigonometrically,  the  calculation  is  thus: 


FIG.  36. — Vector  diagram  of  transformer  with  non-inductive  loading. 
In  triangle  OEEi,  Fig.  34,  writing 

tan  0'  =  ^ 
we  have, 

also, 


OEi2  =  OE2  +  EEi2  -  2  OE  EEi  cos 


EEi  = 

i  =  180  -  tf  +  0i, 


hence, 

E£  =  E2  +  7i2zi2  +  2  #7iZi  cos  (0'  -  00 . 

This  gives  the  secondary  e.m.f.,  EI,  and  therefrom  the  primary 
counter-generated  e.m.f. 

Ei  =  — • 
In  triangle  EOEi  we  have 

sin  E\OE  -T-  sin  E\EO  =  .E^n  -f-  EiO 


ALTERNATING-CURRENT  TRANSFORMER         71 

thus,  writing 

£  E&E  =  0", 
we  have 

sin  0"  -4-  sin  (0'  -  0i)  =  hz  +  Ei, 

wherefrom  we  get 

%  6",  and  £  E1OIl  =  6  =  0,  +  0", 

the  phase  displacement  between  secondary  current  and  secondary 
e.m.f. 


FIG.   37. — Vector  diagram  of  transformer  with  leading  load  current. 

In  triangle  O/oo/o  we  have 


since 


and 


0/02  =  O/oo2  +  /oo/o2  -  2  O/oo/oo/o  COS  O/oo/o, 

£  #i00  =  90°, 
$  O/oo/o  =  90  +  0  +  a, 


J00/0  =  r  =  al 


O/oo  =  IQQ  =  exciting  current, 

calculated  from  the  dimensions  of  the  magnetic  circuit.     Thus 
the  primary  current  is 

J02  =  J002  +  a2/l2  _{_  2  a/ i/oo  sin  (0  +  a). 
In  triangle  O/oo/o  we  have 

sin  /ooO/o  -T-  sin  O/oo/o  =  /oo/o  -^  0/0; 
writing 

this  becomes 

sin  0"0  -^  sin  (0  +  a)  =  a/i  -?-  /0; 

therefrom  we  get  0"0,  and  thus 

£  #'0/0  =  02  =  90°  -  a  -  0"0. 


72       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

In  triangle  OE'EQ  we  have 

OEQ2  =  OE~'2  +  WWo*  -20E'  WW0  cos  OE'  #0; 
writing 

tan  0'0  =  — ' 

7*0 

we  have 

£  OE'E0  =  180°  -  0'  +  02, 

OE'  =  E<  =      > 


thus  the  impressed  e.m.f.  is 

ET    9  ^l2       I        T    9        9       I       *^UO*0  /„    /  „    s 

£V    =    ^2"   +  ^02^02   H COS  (00     —    02). 

In  triangle  OE'EQ 

sin  E'OEo  -T-  sin 
thus,  writing 


we  have 

sin  Q'\  +  sin  (0'0  -  02)  =  /o£o  ^  ^0; 
herefrom  we  get  ^  0"i,  and 

^    00    =    02    +    0"l, 

the  phase  displacement  between  primary  current  and  impressed 
e.m.f. 

As  seen,  the  trigonometric  method  of  transformer  calculation 
is  rather  complicated. 

62.  Somewhat  simpler  is  the  algebraic  method  of  resolving 
into  rectangular  components. 

Considering  first  the  secondary  circuit,  of  current  7i  lagging 
behind  the  terminal  voltage  E  by  angle  0i. 

The  terminal  voltage  E  has  the  components  E  cos  0i  in  phase, 
E  sin  0i  in  quadrature  with  and  ahead  of  the  current  /i. 

The  e.m.f.  consumed  by  resistance  r1}  I-pi,  is  in  phase. 

The  e.m.f.  consumed  by  reactance  x\,  I&i,  is  in  quadrature 
ahead  of  /i. 

Thus  the  secondary  e.m.f.  has  the  components 

E  cos  0i  +  /ifi  in  phase, 

E  sin  0i  +  IiXi  in  quadrature  ahead  of  the  current  /i,  and 
the  total  value, 

Ei  =  V(E  cos  0i  +  /in)2  +  (E  sin  0i  +  /iZi)2, 


ALTERNATING-CURRENT  TRANSFORMER         73 

and  the  tangent  of  the  phase  angle  of  the  secondary  circuit  is 

E  sin  0i  + 


tan  e  = 


E  COS  61  + 


Resolving  all  quantities  into  components  in  phase  and  in 
quadrature  with  the  secondary  e.m.f.  EI,  or  in  horizontal  and 
in  vertical  components,  choosing  the  magnetism  or  mutual  flux 
as  vertical  axis,  and  denoting  the  direction  to  the  right  and 
upward  as  positive,  to  the  left  and  downward  as  negative,  we 
have 

Horizontal  Vertical 

component         component 

Secondary  current,  /i,  —  Ii  cos  0  +  /i  sin  0 

Secondary  e.m.f.,  EI,  —  EI  0 

Primary  counter-generated  e.m.f., 

El  =  ^  -  -1  0 

a  a 

Primary    e.m.f.    consumed    thereby, 

E'  =  -  E,,  +  ^  0 

Primary   load  current,  /'  =  —  a/i,+  al\  cos  6  —  al\  sin  0 
Magnetic  flux,  $>,  0  —  <£ 

Primary  exciting  current,   /Oo,   con- 
sisting of  core  loss  current,  /oo  sin  a 
magnetizing  current,  —  /oo  cos  a. 
hence,  total  primary  current,  J0, 

Horizontal  component  Vertical  component 

all  cos  0i  +  /oo  sin  a  —  (all  sin  0i  +  /oo  cos  a) 

E.m.f.  consumed  by  primary  resistance  r0,  E'Q  =  Ior0  in  phase 
with  /o, 

Horizontal  component  Vertical  component 

r0a/i  cos  0  +  r0/oo  sin  a  —  (r0a/i  sin  0  +  r0/oo  cos  a) 

E.m.f.   consumed  by  primary  reactance  x0,   E0  =  IoXQ,  90° 
ahead  of  /o, 

Horizontal  component  Vertical  component 

X0ali  sin  0  +  Zo/oo  cos  a          +  X0ali  cos  0  +  rr0/oo  sin  a 

Tjl 

E.m.f.    consumed    by    primary    generated    e.m.f.,    E'  =  -—• 
horizontal. 


74       ELEMENTS  OF  ELECTRICAL  ENGINEERING 
The  total  primary  impressed  e.m.f.,  EQ, 

Horizontal  component 
ET 

—  +  ol\  (r0  cos  0  •+•  #o  sin  6)  +  /oo  (TO  sin  a  +  XQ  cos  a). 

Vertical  component 

a/i  (r0  sin  0  —  £0  cos  0)  +  70o  (ro  cos  a  —  XQ  sin  a), 

or  writing  tan  0'o  =  —  > 

?*o 

since 

\A*o2  +  £02  =  ZQ,  sin  0'0  =  —  ,  and  cos  0'0  =  -°' 

20  ZQ 


Substituting  this  value,  the  horizontal  component  of  E0  is 
•pi 

—  +  a20/i  cos  (0  —  0'0)  +  2o/oo  sin  (a  +  0r0)  ; 

the  vertical  component  of  E0  is 

azo/i  sin  (0  —  0'0)  +  20/oo  cos  (a  +  0'0), 
and,  the  total  primary  impressed  e.m.f.  is 

£o=\/r—  +azo/icos(0—  0'o)+*oloosin(a+0'o)]>+ 


2azo/oo  .  .  ,  o*zo2/i2  ,  a22o2/oo2     .  2a"zo2/i/oo  . 

Mn(a+g/o)+—        -  +—      —  +  - 


Combining  the  two  components,  the  total  primary  current  is 
0  +  /oo  sin  a)2  +  (a/i  sin  0  +  /oo  cos  a) 


Since  the  tangent  of  the  phase  angle  is  the  ratio  of  vertical 
component  to  horizontal  component,  we  have,  primary  e.m.f. 
phase, 

_         azp/i  sin  (0  —  0'0)  -f  ZO/OQ  cos  («  +  0'o) 
tan  0    —  -j=, 

—  +  020/1  cos  (0  -  0'0)  +  20/oo  sin  (a  -  0'0) 


primary  current  phase, 
„ 

and  lag  of  primary  current  behind  impressed  e.m.f., 


„  _  ali  sin  0  -f  /oo  cos  a 
~  ali  cos  0  +  /oo  sin  a 


ALTERNATING-CURRENT  TRANSFORMER        75 


EXAMPLES 

63.  (1)  In  a  20-kw.  transformer  the  ratio  of  turns  is  20  -f-  1, 
and  100  volts  is  produced  at  the  secondary  terminals  at  full 
load.  What  is  the  primary  current  at  full  load,  and  the  regu- 
lation, that  is,  the  rise  of  secondary  voltage  from  full  load  to  no 
load,  at  constant  primary  voltage,  and  what  is  this  primary 
voltage? 

(a)  at  non-inductive  secondary  load, 

(6)  with  60  degrees  time  lag  in  the  external  secondary  circuit, 

(c)  with  60  degrees  time  lead  in  the  external  secondary  circuit. 

The  exciting  current  is  0.5  amp.,  the  core  loss  600  watts, 
the  primary  resistance  2  ohms,  the  primary  reactance  5  ohms, 
the  secondary  resistance  0.004  ohm,  the  secondary  reactance 
0.01  ohm. 

Exciting  current  and  core  loss  may  be  assumed  as  constant. 

600  watts  at  2000  volts  gives  0.3  amp.  core  loss  current, 
hence  V0.52  —  32  =  0.4  amp.  magnetizing  current. 

We  have  thus 


r0  =  2 

XQ   —   5 


=  0.004 
=  0.01 


/oo  cos  a  =  0.3 
/oo  sin  a  =  0.4 
/oo  =  0.5 


0.05 


1.  Secondary  current  as  horizontal  axis: 


Non-inductive, 
0i  =0 

01  -*+W 

Le*d, 

0l  =   -  60° 

Hor. 

Vert. 

Hor. 

Vert. 

Hor. 

Vert. 

Secondary  current,  /i.  . 
Secondary  terminal 
voltage,  E. 

200 

100 

0.8 
0 
100.8 

0 

% 

0 
0 
+2.0 
+2.0 

200 

50 
0.8 

0 
50.8 

0 

+86.6 

° 

+  2.0 

+88.6 

200 

50 
0.8 
0 
50.8 

0 

-86.6 
0 
+  2.0 
-84.6 

Resistance  voltage,  I\T\. 
Reactance  voltage,  I\xi. 
Secondary  e.m.f  .,  E\... 

Secondary  e.m.f.,  total 
tan  0 

100.80 
+0.0198 

+  1.1° 

102.13 

+  1.745 
+60.2° 

98.68 
-   1.665 
-59.0° 

76       ELEMENTS  OF  ELECTRICAL  ENGINEERING 
2.  Magnetic  flux  as  vertical  axis: 


Non-inductive, 
61  =  0 

Lag, 

0!  =   +  60° 

Lead, 

0i  =-  60° 

Hor. 

Vert. 

Hor. 

Vert. 

Hor. 

Vert. 

Secondary  gen- 
erated    e.m.f., 
E  

-100.80 
-200 

+  10 

0.3 
+  10.3 

20.6 
3.0 

2016 
2039.6 

0 

+  4 

+  0.2 
0.4 
-  0.6 

1.2 
+51.3 

0 

+50.1 

-102.13 
-  99.4 

+     4.97 
0.3 
+     5.27 

10.54 
45.20 

2042.6 
2098.34 

0 

-172.8 

-     8.64 
-     0.4 
-     9.04 

-   18.08 
+  26.35 

0 

+     8.27 

-  98.68 
-103 

+     5.15 
0.3 
+     5.45 

10.90 
-  40.85 

1973.6 
1943.65 

0 
-171.4 

+  8.57 
-  0.4 
+  8.17 

+  16.34 

+27.25 

0 

+43.59 

Secondary   cur- 
rent, 1  1  
Primary       load 
current,     /'  = 
-a/i 

Primary    excit- 
ing current,  70o 
Total     primary 
current,  /o  .  .  .  . 
Primary   resist- 
ance,   voltage, 
/oPo  

Primary    react- 
ance,   voltage, 
Iox0  
E.m.f.  consum- 
ed by  primary 
counter  e.m.f., 
-El 

a 
Total     primary 
impressed  e.m.f., 
E°    

Hence, 


Non-inductive, 

0i  =  0 

Lag, 

0i  =   +  60° 

Lead, 

0!    =     -    60° 

Resultant  Eo                               .... 

2040  1 

2098  3 

1944  2 

Resultant  /o 

10  32 

10  47 

9  82 

Phase  of  E0       

-1.4° 

-  0  2° 

—   1  2° 

Phase  of  /o 

+3.3° 

+59  8° 

—56  3° 

Primary  lag   60 

+4.7° 

+60  0° 

—  55  1° 

W 

Rpsrulation  

1  02005 

1  04915 

0  9721 

2000'  ' 
Drop  of  voltage,  per  cent 

2  005 

4  915 

—  2  79 

Change  of  phase,  do  —  0i  

4.7° 

0 

4.9° 

RECTANGULAR  COORDINATES  77 

14.  RECTANGULAR  COORDINATES 

64.  The  vector  diagram  of  sine  waves  gives  the  best  insight 
into  the  mutual  relations  of  alternating  currents  and  e.m.fs. 

For  numerical  calculation  from  the  vector  diagram  either  the 
trigonometric  method  or  the  method  of  rectangular  components 
is  used. 

The  method  of  rectangular  components,  as  explained  in  the 
above  paragraphs,  is  usually  simpler  and  more  convenient  than 
the  trigonometric  method. 

In  the  method  of  rectangular  components  it  is  desirable  to 
distinguish  the  two  components  from  each  other  and  from  the 
resultant  or  total  value  by  their  notation. 

To  distinguish  the  components  from  the  resultant,  small 
letters  are  used  for  the  components,  capitals  for  the  resultant. 
Thus  in  the  transformer  diagram  of  Section  13  the  secondary 
current  I\  has  the  horizontal  component  ii  =  —  I\  cos  0i,  and 
the  vertical  component  i'\  —  +  I\  sin  0\. 

To  distinguish  horizontal  and  vertical  components  from  each 
other,  either  different  types  of  letters  can  be  used,  or  indices,  or 
a  prefix  or  coefficient. 

Different  types  of  letters  are  inconvenient,  indices  distinguish- 
ing the  components  undesirable,  since  indices  are  reserved  for 
distinguishing  different  e.m.fs.,  currents,  etc.,  from  each  other. 

Thus  the  most  convenient  way  is  the  addition  of  a  prefix  or 
coefficient  to  one  of  the  components,  and  as  such  the  letter  j  is 
commonly  used  with  the  vertical  component. 

Thus  the  secondary  current  in  the  transformer  diagram, 
Section  13,  can  be  written 

i\  +  ji*  =  Ii  cos  0i  +  jli  sin  0i.  (1) 

This  method  offers  the  further  advantage  that  the  two  com- 
ponents can  be  written  side  by  side,  with  the  plus  sign  between 
them,  since  the  addition  of  the  prefix  j  distinguishes  the  value 
jit  or  jli  sin  0i  as  vertical  component  from  the  horizontal  com- 
ponent i\  or  1 1  cos  0i. 

1 1  =  ii  +  ji*  (2) 

thus  means  that  I\  consists  of  a  horizontal  component  i\  and  a 
vertical  component  iz,  and  the  plus  sign  signifies  that  i\  and  iz  are 
combined  by  the  parallelogram  of  sine  waves. 


78       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

The  secondary  e.m.f.  of  the  transformer  in  Section  13,  Fig.  34, 
is  written  in  this  manner,  E\  =  —  ei,  that  is,  it  has  the  hori- 
zontal component  —  e\  and  no  vertical  component. 

The  primary  generated  e.m.f.  is 

E'=='  00 


and  the  e.m.f.  consumed  thereby 

E'  =  +  ei-  w 

The  secondary  current  is 

where 

ii  =  Ii  cos  0i,     iz  =  Ii  sin  0i,  (6) 

and  the  primary  load  current  corresponding  thereto  is 

I'  =  -  aii  =  aii  -  jaiz.  (7) 

The  primary  exciting  current, 

Joo  =  h  -  jg,  (8) 

where  h  =  J0o  sin  a  is  the  hysteresis  current,  g  =  I0o  cos  a  the 
reactive  magnetizing  current. 
Thus  the  total  primary  current  is 

J0  =  I'  +  J00  =  (aii  +  h)  -j  (aiz  +  g).  (9) 

The  e.m.f.  consumed  by  primary  resistance  rQ  is 

r0Jo  =  TQ  (aii  +  h)  -  jr0  (aiz  +  0).  (10) 

The  horizontal  component  of  primary  current  (aii  +  h)  gives 
as  e.m.f.  consumed  by  reactance  XQ  a  negative  vertical  com- 
ponent, denoted  by  JXQ  (aii  +  h).  The  vertical  component  of 
primary  current  j  (aiz  +  g)  gives  as  e.m.f.  consumed  by  react- 
ance XQ  a  positive  horizontal  component,  denoted  by  XQ  (aiz  +  (/)• 

Thus  the  total  e.m.f.  consumed  by  primary  reactance  XQ  is 

XQ  (aiz  +  g)  +  jxQ  (aii  +  h),  (11) 

and  the  total  e.m.f.  consumed  by  primary  impedance  is 
r0  (aii  +  A)  +  x0  (aiz  +  g)  -  j[rQ  (aiz  +  g)  -  XQ  (aii  +  h)].  (12) 


RECTANGULAR  COORDINATES  79 

Thus,  to  get  from  the  current  the  e.m.f.  consumed  in  react- 
ance XQ  by  the  horizontal  component  of  current,  the  coefficient 
j  has  to  be  added;  in  the  vertical  component  the  coefficient  —  j 
omitted;  or,  we  can  say  the  reactance  is  denoted  by  jxQ  for  the 

horizontal  and  by r-  for  the  vertical  component  of  current. 

In  other  words,  if  7  =  i  —  ji'  is  a  current,  x  the  reactance  of  its 
circuit,  the  e.m.f.  consumed  by  the  reactance  is 

jxi  H-  xi'  =  xi'  +  jxi. 

65.  If  instead  of  omitting  —  j  in  deriving  the  reactance  e.m.f. 
for  the  vertical  component  of  current  we  would  add  j  also  (as 
done  when  deriving  the  reactance  e.m  f.  for  the  horizontal 
component  of  current),  we  get  the  reactance  e.m.f. 

jxi  —  fxi', 
which  gives  the  correct  value  jxi  +  xi',  if 

f  =  -  1;  (13) 

that  is,  we  can  say,  in  deriving  the  e.m.f.  consumed  by  reactance, 
x,  from  the  current,  we  multiply  the  current  by  jx,  and  substitute 

By  defining,  and  substituting,  j2  =  —  1,  jx  can  thus  be  called 
the  reactance  in  the  representation  in  rectangular  coordinates 
and  r  -+-  jx  the  impedance. 

The  primary  impedance  voltage  of  the  transformer  in  the 
preceding  could  thus  be  derived  directly  by  multiplying  the 
current, 

/o  =  (aii  +  h)  -  j  (aii  +  g),  (9) 

by  the  impedance, 

Z0  =  r0  -f  jxQ, 
which  gives 


E'o  =  Zo/o  =  (r0  +  jx<>)  [(aii  +  h)  -  j  (ai2  +  g)] 

=  r0  (aii  +  h)  -  jrQ  (ai2  +  g)  +  jxQ  (aii  +  h)  -  j2xQ  (ai2  -f  g), 
and  substituting  j2  =  —  1, 

E'o  =  [r0  (aii  +  h)  +  XQ  (ai2  +  g)]  -  j  [r0  (ai2  +  g)  -  XQ  (aii  +  h)], 

(14) 


80       ELEMENTS  OF  ELECTRICAL  ENGINEERING 


and  the  total  primary  impressed  e.m.f.  is  thus 

EQ  =  E     -["•  E  o 

=  [^  +  r0  (ail  +  h)  +  x0  (<M2+s)J  -j  [r0  (era's  +  ff)  -z0  (ai  +  K)]. 

(15) 
66.  Such  an  expression  in  rectangular  coordinates  as 

I  =  i  +  jV  (16) 

represents  not  only  the  current  strength  but  also  its  phase. 

It  means,  in_Fig.  38,  that  the 
total  current  01  has  the  two  rect- 
angular components,  the  horizontal 
component  I  cos  6  =  i  and  the 
vertical  component  /  sin  6  =  i'. 

Thus, 


tan  0  =  - 


(17) 


FIG.  38. — Magnitude  and  phase 
in  rectangular  coordinates. 


that  is,  the  tangent  function  of  the 
phase  angle  is  the  vertical  compo- 
nent divided  by  the  horizontal  com- 

ponent, or  the  term  with  prefix  j  divided  by  the  term  without  j. 
The  total  current  intensity  is  obviously 

I  =  V>  +  i'2>  (18) 

The  capital  letter  I  in  the  symbolic  expression  /  =  i  +  jif 

thus  represents  more  than  the  /  used  in  the  preceding  for  total 
current,  etc.,  and  gives  not  only  the  intensity  but  also  the  phase. 
It  is  thus  necessary  to  distinguish  by  the  type  of  the  latter  the 
capital  letters  denoting  the  resultant  current  in  symbolic  expres- 
sion (that  is,  giving  intensity  and  phase)  from  the  capital  letters 
giving  merely  the  intensity  regardless  of  phase;  that  is, 

I  = 


denotes  a  current  of  intensity 

/  = 

and  phase 


tan  0  =  — . 
^ 


RECTANGULAR  COORDINATES  81 

In  the  following,  dotted  italics  wfll  be  used  for  the  symbolic 
expressions  and  plain  italics  for  the  absolute  values  of  alternating 
waves. 

In  the  same  way  z  =  \/r2  +  x2  is  denoted  in  symbolic  repre- 
sentation of  its  rectangular  components  by 

Z  =  r  +  jx.  (91) 

When  using  the  symbolic  expression  of  rectangular  coordinates 
it  is  necessary  ultimately  to  reduce  to  common  expressions.  ; 

Thus  in  the  above  discussed  transformer  the  symbolic  expres- 
sion of  primary  impressed  e.m.f. 

EQ  =  |j^  +  rQ  (aii  +  h)  +  X0(ai2  +g)  J  -j  [r0(ai2+0)  -z0(a*'i+/i)J 

(15) 
means  that  the  primary  impressed  e.m.f.  has  the  intensity 


(ai'2+flf)J 


(20) 
and  the  phase 


tan  0o  = 


-1  +  r0  (aii  +  h)  +  X0  (aiz  +  flf) 

This  symbolism  of  rectangular  components  is  the  quickest 
and  simplest  method  of  dealing  with  alternating-current  phenom- 
ena, and  is  in  many  more  complicated  cases  the  only  method 
which  can  solve  the  problem  at  all,  and  therefore  the  reader 
must  become  fully  familiar  with  this  method. 

EXAMPLES 

67.  (1)  In  a  20-kw.  transformer  the  ratio  of  turns  is  20  :  1, 
and  100  volts  are  required  at  the  secondary  terminals  at  full 
load.  What  is  the  primary  current,  the  primary  impressed 
e.m.f.,  and  the  primary  lag, 

(a)  at  non-inductive  load,  0i  =  0; 

(6)  with  0i  =  60  degrees  time  lag  in  the  external  secondary 
circuit; 

(c)  with  61  =  —  60  degrees  time  lead  in  the  external  secondary 
circuit? 


82       ELEMENTS  OF  ELECTRICAL  ENGINEERING 


i    i 

«!s 
32 


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84       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

The  exciting  current  is  /'00  =  0.3  —  0.4  j  amp.  at  e  =  2000 
volts  impressed,  or  rather,  primary  counter-generated  e.m.f. 

The  primary  impedance,  Z0  =  2  -f  5  j  ohms. 

The  secondary  impedance,  Z\  =  0.004  +  0.01  j  ohm. 

We  have,  in  symbolic  expression,  choosing  the  secondary 
current  /i  as  real  axis,  the  results  calculated  in  tabulated  form 
on  page  82. 

68.  (2)  eQ  =  2000    volts    are    impressed    upon    the    primary 
circuit  of  a  transformer  of  ratio  of  turns  20:1.     The  primary 
impedance  is  ZQ  =  2  -f  5  j,   the  secondary  impedance,   Zi  = 
0.004  +  0.01  j,    and   the   exciting   current   at   er  =  2000   volts 
counter-generated  e.m.f.  is  70o  =  0.3  —  0.4  j;  thus  the  exciting 

admittance,  Y  =  ^  =  (0.15  -  0.2  j)10~3. 

6 

What  is  the  secondary  current  and  secondary  terminal  voltage 
and  the  primary  current  if  the  total  impedance  of  the  secondary 
circuit  (internal  impedance  plus  external  load)  consists  of 

(a)  resistance, 

Z  =  r  =  0.5  —  non-inductive  circuit. 
(6)  impedance, 

Z  =  r  +  jx  =  0.3  +  0.4  j  —  inductive  circuit. 
(c)   impedance, 

Z  =  r  -\-  jx  =  0.3  —  0.4  j  —  anti-inductive  circuit. 
Let  e  =  secondary  e.m.f., 

assumed  as  real  axis  in  symbolic  expression,  and  carrying  out 
the  calculation  in  tabulated  form,  on  page  83. 

69.  (3)  A    transmission     line    of    impedance    Z  =  r  -}-  jx  = 
20  +  50  j  ohms  feeds  a  receiving  circuit.     At  the  receiving  end  an 
apparatus    is    connected    which    produces    reactive    lagging   or 
leading  currents  at  will  (synchronous  machine)  ;  12,000  volts  are 
impressed    upon    the    line.     How    much    lagging    and    leading 
currents  respectively  must  be  produced  at  the  receiving  end  of 
the  line  to  get   10,000  volts  (a)   at   no  load,  (6)    at   50  amp. 
power    current    as    load,   (c)    at    100   amp.    power    current   as 
load? 

Let  e  =  10,000  =  e.m.f.  received  at  end  of  line,  ii  =  power 
current,  and  i%  =  reactive  lagging  current;  then 


total  line  current. 


LOAD  CHARACTERISTIC  OF  TRANSMISSION  LINE  85 

The  voltage  at  the  generator  end  of  the  line  is  then 
E0  =  e  +  ZI 

=  e  +  (r  +  jx)  (ii  —  jiz) 
=  (e  +  rii  +  xi2)  —  j  (n'2  —  xii) 
=  (10,000  +  20  ti  +  50 1*2)  -  j  (20  za  -  50t'i); 
or,  reduced, 


•f-ni  +  zi2)2  +  (ri2  - 
thus,  since  E0  =  12,000, 

12,000  =  V(10,000  +  20*i  +  50^)2  +  (20 iz  -  50^i)2.     . 
(a)  At  no  load  i\  =  0,  and 
12,000  = 


hence, 

i2  =  -j-  39.5  amp.,  reactive  lagging  current,  I  =  —  39.5 ./. 

(6)  At  half  load  ii  =  50,  and 


12,000  =  VuMXX)  +  50i2)2  +  (20z2  -  2500)2;- 
hence, 

is  =  +  16  amp.,  lagging  current,/  =  50  —  16  j. 

(c)  At  full  load  ii  =  100,  and 


12,000  =  V(12,000  +  50i2)2  +  (20 i  -  5000)2; 
hence, 

i2  =  -  27.13  amp.,  leading  current,  I  =  100  +  27.13  j. 

15.  LOAD    CHARACTERISTIC    OF   TRANSMISSION   LINE 

70.  The  load  characteristic  of  a  transmission  line  is  the  curve 
of  volts  and  watts  at  the  receiving  end  of  the  line  as  function  of 
the  amperes,  and  at  constant  e.m.f .  impressed  upon  the  generator 
end  of  the  line. 

Let  r  =  resistance,  x  =  reactance  of  the  line.  Its  impedance 
z  =  -y/r2  +  x2  can  be  denoted  symbolically  by 

Z  =  r  +  jx. 

Let  EQ  =  e.m.f.  impressed  upon  the  line. 
Choosing  the  e.m.f.  at  the  end  of  the  line  as  horizontal  com- 
ponent in  the  vector  diagram,  it  can  be  denoted  by  E  =  e. 


86       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

At  non-inductive  load  the  line  current  is  in  phase  with  the 
e.m.f.  e,  thus  denoted  by  7  =  i. 

The  e.m.f.  consumed  by  the  line  impedance  Z  —  r  +  jx  is 
E!  =  ZI  =  (r  +  jx)  i 

=  ri+jxi.  (1) 

Thus  the  impressed  voltage, 

'  Eo  =  E  +  Ei  =  e  +  ri  +  ja».  (2) 

or,  reduced, 

#o  =  V(e  +  n)2  +  z2*2,  (3) 

and 


_ 

6  =      ^o2  -  z2*2  -  n,  the  e.m.f.  (4) 


p  =  d  =  i  V-Eo2  -  x2i2  -  ri2,  (5) 

the  power  received  at  end  of  the  line. 

The  curve  of  e.m.f.  e  is  an  arc  of  an  ellipse. 

With   open  circuit  i  =  0,  e  =  E0  and  P  =  0,   as  is   to   be 
expected. 

At  short  circuit,  e  =  0,  0  =  \/#o2  —  xzi2  —  ri,  and 

°          ;  (6) 


X' 

that  is,  the  maximum  line  current  which  can  be  established  with 
a  non-inductive  receiver  circuit  and  negligible  line  capacity. 
71.  The  condition  of  maximum  'power  delivered  over  the  line 

'•      i|       f-*  on 

that  is, 


substituting  (3): 

'!  V#o2  -  x*i*  =  e  +  ri, 

and  expanding,  gives 

e*  =  (r2  +  x2)  i2  (8) 

=  z2i2; 


hence, 


e  —  zi,    and  -  =  z.  (9) 


-T-  =  7*1  is  the  resistance  or  effective  resistance  of  the  receiving 
circuit;  that  is,  the  maximum  power  is  delivered  into  a  non- 


LOAD  CHARACTERISTIC  OF  TRANSMISSION  LINE  87 

inductive  receiving  circuit  over  an  inductive  line  upon  which  is 
impressed  a  constant  e.m.f.,  if  the  resistance  of  the  receiving 
circuit  equals  the  impedance  of  the  line,  TI  =  z. 
In  this  case  the  total  impedance  of  the  system  is 

Z0  =  Z  +  n  =  r  +  z  +  jx,  (10) 

or,  

zo  =  V(r  +  z)2  +  z2.  (11) 

Thus  the  current  is 


*o       V(r  +  z)2  +  x2 

and  the  power  transmitted  is 

Eo2z 


(r 


that  is,  the  maximum  power  which  can  be  transmitted  over  a 
line  of  resistance  r  and  reactance  x  is  the  square  of  the  impressed 
e.m.f.  divided  by  twice  the  sum  of  resistance  and  impedance  of 
the  line. 

At  x  =  0,  this  gives  the  common  formula, 


Inductive  Load 

72.  With  an  inductive  receiving  circuit  of  lag  angle  6,  or 
power-factor  p  =  cos  8,  and  inductance  factor  q  =  sin  6,  at 
e.m.f.  E  =  e  at  receiving  circuit,  the  current  is  denoted  by 

I  =  I(p-jq);  (15) 

thus  the  e.m.f.  consumed  by  the  line  impedance  Z  =  r  -f  jx  is 
E!  =  ZI  =  I  (p  -jq)(r+jx) 
=  I  [(rp  +  xq)  -  j  (rq  -  xp)], 

and  the  generator  voltage  is 

Eo  =  E  +  #1 

=  [e  +  /  (rp  +  sg)].  -  jl  (rq  -  xp);  (16) 


88       ELEMENTS  OF  ELECTRICAL  ENGINEERING 
or,  reduced, 


#o  =  V+  7  (rp  +  xq)}2  +  P  (rq  -  xp)*,  (17) 

and 

e  =  \/EQ*-  P(rq-xp)2  -  I  (rp  +  xq).          (18) 

The  power  received  is  the  e.m.f.  times  the  power  component 
of  the  current;  thus 

P  =  elp     __ 

=  Ip  \/Eo*-  P(rq-xp)*  -  Pp  (rp  +  xq).         (19) 

The  curve  of  e.m.f.,  e,  as  function  of  the  current  I  is  again  an 
arc  of  an  ellipse. 

At  short  circuit  e  =  0;  thus,  substituted, 

/-£•  (20) 

the  same  value  as  with  non-inductive  load,  as  is  obvious. 

73.  The  condition  of  maximum  output  delivered  over  the 
line  is 


that  is,  differentiated, 


V#o2  -I2(rq-xp)2  =  e  +  I  (rp  +  xq);  (22) 

substituting  and  expanding, 


e  =  Iz] 
or 

y  =  z.  (23) 

Zi  =  -j  is  the  impedance  of  the  receiving  circuit;  that  is,  the 

power  received  in  an  inductive  circuit  over  an  inductive  line  is 
a  maximum  if  the  impedance  of  the  receiving  circuit,  z\y  equals 
the  impedance  of  the  line,  z. 

In  this  case  the  impedance  of  the  receiving  circuit  is 

Zi  =  z(p  +jq),  (24) 

and  the  total  impendance  of  the  system  is 

ZQ  =   Z  -{-  Zi 

=  r  +  jx  +  z  (p  +  jq) 


LOAD  CHARACTERISTIC  OF  TRANSMISSION  LINE  89 
Thus,  the  current  is 


/i  = 


and  the  power  is 


V(r-f 


2  (z  +  rp  +  xq) 


EXAMPLES 


(25) 


(26) 


74.  (1)   12,000  volts  are  impressed  upon  a  transmission  line 
of  impedance  Z  =  r  +  jx  =  20  +  50  j.     How  do  the  voltage 


\ 


\ 


\ 


V 


son 


mo 


VOLTS 

11000 


9000 


7000 


4000 


20    40    60    80    100    120    140    160   .180   200   220 

FIG.  39. — Non-reactive  load  characteristic^  of  a  transmission  line.     Con- 
stant impressed  e.m.f. 

and  the  output  in  the  receiving  circuit  vary  with  the  current 
with  non-inductive  load? 

Let  e  =  voltage  at  the  receiving  end  of  the  line,  i  =  current: 
thus  =  ei  —  power  received.  The  voltage  impressed  upon  the 
line  is  then 

Eo  =  e  +  Zi 

=  e  +ri  +  jxi; 


90       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

or,  reduced, 

Eo  =  V( 
Since  EQ  =  12,000, 


12,000  =  V(e  +  n)2  +  xH*   =  V(e  +  20  i)*  +  2500  i\ 

e  =  V12,0002  -  x2i2   -  ri  =  Vl2,0002  -  2500^  -  20 

The  maximum  current  for  e  =  0  is 


thus, 


0  =  V12,0002  -  2500  i2  -  20  i\ 


i  =  223. 

Substituting  for  i  gives  the  values  plotted  in  Fig.  39. 


i 

e 

p  =  ei 

0 

12,000 

0 

20 

11,500 

230  X  103 

40 

11,000 

440  X  103 

60 

10,400 

624  X  103 

80 

9,700 

776  X  103 

100 

8,900 

890  X  103 

120 

8,000 

960  X  103 

140 

6,940 

971  X  103 

160 

5,750 

920  X  103 

180 

4,340 

784  X  103 

200 

2,630 

526  X  103 

220 

400 

88  X  103 

223 

0 

0 

16.  PHASE  CONTROL  OF  TRANSMISSION  LINES 

76.  If  in  the  receiving  circuit  of  an  inductive  transmission 
line  the  phase  relation  can  be  changed,  the  drop  of  voltage  in 
the  line  can  be  maintained  constant  at  varying  loads  or  even 
decreased  with  increasing  load;  that  is,  at  constant  generator 
voltage  the  transmission  can  be  compounded  for  constant  voltage 
at  the  receiving  end,  or  even  over-compounded  for  a  voltage 
increasing  with  the  load. 

1.  Compounding  of  Transmission  Lines  for  Constant  Voltage 

Let  r  =  resistance,  x  =  reactance  of  the  transmission  line, 
CQ  =  voltage  impressed  upon  the  beginning  of  the  line,  e  =  vol- 
tage received  at  the  end  of  end  line. 


PHASE  CONTROL  OF  TRANSMISSION  LINES     91 

Let  i  =  power  current  in  the  receiving  circuit;  that  is,  P  — 
ei  =  transmitted  power,  and  ii  =  reactive  current  produced 
in  the  system  for  controlling  the  voltage.  i\  shall  be  considered 
positive  as  lagging,  negative  as  leading  current. 

Then  the  total  current,  in  symbolic  representation,  is 

/  =  i  -  jii; 

the  line  impedance  is 

Z  =  r  +  jx, 

and  thus  the  e.m.f.  consumed  by  the  line  impedance  is 

Ei  =  ZI  =  (r  +  jx)  (i  -  jii) 
=  ri  +  jrii  +  jxi  -  J2xii; 

and  substituting  f  —  —  1, 

Ei  =  (ri  +  xii)  -  j  (rii  -  xi). 
Hence  the  voltage  impressed  upon  the  line 

Eo  =  e  4-  Ei 

=  (e  +  ri  +  xii)  -  j  (rii  -  xi)  ;  (1) 

or,  reduced, 


_ 

eo  =  V(e  +  ri  +  xii)*  +  (n\  -  xi)*.  (2) 

If  in  this  equation  e  and  eQ  are  constant,  ii,  the  reactive  com- 
ponent of  the  current,  is  given  as  a  function  of  the  power  com- 
ponent current  i  and  thus  of  the  load  ei. 

Hence  either  eQ  and  e  can  be  chosen,  or  one  of  the  e.m.fs.  eQ 
or  e  and  the  reactive  current  ii  corresponding  to  a  given  power 
current  i. 

76.  If  ii  =  0  with  i  =  0,  and  e  is  assumed  as  given,  eQ  =  e. 
Thus,  _ 

e  =  V(e  +  ri  +  xi^  +  (rii  -  xi)*; 
2  e  (ri  +  xii)  +  (r2  +  x2)  (i2  +  ii2)  =  0. 

From  this  equation  it  follows  that 


ex  ±  Ve2x2  -  2  eriz2  -  i2z*  /ox 

*i  -  -  -  — i~ 

Thus,  the  reactive  current  ii  must  be  varied  by  this  equation 
to  maintain  constant  voltage  e  =  eo  irrespective  of  the  load  ei. 

As  seen,  in  this  equation,  ii  must  always  be  negative,  that  is, 
the  current  leading. 


92       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

ii  becomes  impossible  if  the  term   under  the  square  root 
becomes  negative,  that  is,  at  the  value 

e2x2  -  2  eriz2  -  i2z4  =  0; 

1       ,'  f-^4  -  (4) 

At  this  point  the  power  transmitted  is 


This  is  the  maximum  power  which  can  be  transmitted  with- 

Q  (g  _  Y\ 

out  drop  of  voltage  in  the  line,  with  a  power  current  i  =  -  ^  —  . 

The  reactive  current  corresponding  hereto,  since  the  square 
root  becomes  zero,  is 

III  ,-.....'.;    ti  =  5[;  .  .  .  (6) 

thus  the  ratio  of  reactive  to  power  current,  or  the  tangent  of 
the  phase  angle  of  the  receiving  circuit,  is 


A  larger  amount  of  power  is  transmitted  if  e0  is  chosen  >  e, 
a  smaller  amount  of  power  if  eQ  <  e. 

In  the  latter  case  ii  is  always  leading;  in  the  former  case  i\ 
is  lagging  at  no  load,  becomes  zero  at  some  intermediate  load, 
and  leading  at  higher  load. 

77.  If  the  line  impedance  Z  —  r  +  fa  and  the  received  voltage 
e  is  given,  and  the  power  current  ^o  at  which  the  reactive  current 
shall  be  zero,  the  voltage  at  the  generator  end  of  the  line  is 
determined  hereby  from  the  equation  (2)  : 


eQ  =  V(e  -f  ri  +  xii)2  +  (ri{  -  xi)2, 
by  substituting  i\  =  0,  i  =  z'o, 


Substituting  this  value  in  the  general  equation  (2)  : 


e0  =  V(e  +  ri 
gives 

(e  +  n0)2  +  zV  =  (e  +  ri  +  xitf  +  (rii  -  xi)2          (9) 
as  equation  between  i  and  i\. 


PHASE  CONTROL  OF  TRANSMISSION  LINES     93 

If  at  constant  generator  voltage  e0: 
at  no  load, 

i  =  0,  e  =  e0,  i\  =  i'o, 


and  at  the  load, 


(10) 


i  =  i  o,  6  =  BO,  i\  =  0 
it  is,  substituted: 
no  load,  

load  io, 

Thus, 

(eo  +  £fc'o)  +  #V|  =  (eo  +  n'o)2  +  x2t*o2; 
or,  expanded, 

iV(r2  +  x2)  +  2  i'0  xe0  =  io2  (r2  +  x2)  +  2  iQre0.         (13) 
This  equation  gives  i'o  as  function  of  io,  e0,  r,  x. 
If  now  the  reactive  current  i\  varies  as  linear  function  of  the 
power  current  i,  as  in  case  of  compounding  by  rotary  converter 
with  shunt  and  series  field,  it  is 


Substituting  this  value  in  the  general  equation 

(eo  +  n0)2  +  *V  =  (e  +  ri  +  a»i)»  +  (rii  -  xz)2 

gives  e  as  function  of  i;  that  is,  gives  the  voltage  at  the  receiving 
end  as  function  of  the  load,  at  constant  voltage  60  at  the  gener- 
ating end,  and  e  =  eo  for  no  load, 

i  =  0,  ii  =  i'o, 
and  e  =  eQ  for  the  load, 

i  =  io,  ii  =  0. 

Between  i  =  0  and  i  =  io,  e  >  eo,  and  the  current  is  lagging. 

Above  i  =  io,  e  <  eQ,  and  the  current  is  leading. 

By  the  reaction  of  the  variation  of  e  from  eo  upon  the  receiving 
apparatus  producing  reactive  current  z'i,  and  by  magnetic  satura- 
tion in  the  receiving  apparatus,  the  deviation  of  e  from  eo  is 
reduced,  that  is,  the  regulation  improved. 

2.  Over-compounding  of  Transmission  Lines 

78.  The  impressed  voltage  at  the  generator  end  of  the  line 
was  found  in  the  preceding, 

eo  =  V(e  4-  ri  +  a»i)a  +  (rii  -  xi)2.  (2) 


94       ELEMENTS  OF  ELECTRICAL  ENGINEERING 

If  the  voltage  at  the  end  of  the  line  e  shall  rise  proportionally 
to  the  power  current  i,  then 

e  =  e\  +  ai',  (15) 

thus, 


eo  =  V[ej.  +  (a  +  r)  t  +  a»i]»  +  (n'i  -  **)2, 
and  herefrom  in  the  same  way  as  in  the  preceding  we  get  the 
characteristic  curve  of  the  transmission. 

If  eo  —  e\t  i\  =  0  at  no  load,  and  is  leading  at  load.  If 
£o  <  ei,  ii  is  always  leading,  the  maximum  output  is  less  than 
before. 

If  eo  >  ei,  i\  is  lagging  at  no  load,  becomes  zero  at  some  inter- 
mediate load,  and  leading  at  higher  load.  The  maximum  output 
is  greater  than  at  e0  =  e\. 

The  greater  a,  the  less  is  the  maximum  output  at  the  same 
GO  and  €\. 

The  greater  eo,  the  greater  is  the  maximum  output  at  the 
same  e\  and  a,  but  the  greater  at  the  same  time  the  lagging  current 
(or  less  the  leading  current)  at  no  load. 

EXAMPLES 

79.  (1)  A  constant  voltage  of  e0  is  impressed  upon  a  trans- 
mission line  of  impedance  Z  =  r  +  jx  =  10  +  20  j.  The  vol- 
tage at  the  receiving  end  shall  be  10,000  at  no  load  as  well  as  at 
full  load  of  75  amp.  power  current.  The  reactive  current  in 
the  receiving  circuit  is  raised  proportionally  to  the  load,  so  as 
to  be  lagging  at  no  load,  zero  at  full  load  or  75  amp.,  and  lead- 
ing beyond  this.  What  voltage  e0  has  to  be  impressed  upon  the 
line,  and  what  is  the  voltage  e  at  the  receiving  end  at  J£,  %,  and 
1J£  load? 

Let  J  =  ii  —  jiz  =  current,  E  =  e  voltage  in  receiving  circuit. 

The  generator  voltage  is  then 

Eo  =  e  +  ZI 

=  e  +  (r  +  jx)  (ii  -  ji2) 

=  (e  +  rii  +  xiz)  —  j  (n'2  —  xii) 

=  (e  +  10  n  +  20  i2)  -  j  (10  t,  -  20  t'O; 
or,  reduced, 

eQ*  =  (e  +  n'j  +  xizy  4.  (n'2  _  ^2. 

=  (e  +  10  ii  +  20  tj)J  +  (10  t,  -  20  *i)2. 
When 

t'i  =  75,  t,  =  0,  e  =  10,000; 


PHASE  CONTROL  OF  TRANSMISSION  LINES     95 

substituting  these  values, 

e02  =  10,7502  +  15002  =  117.81  X  106; 

eo  =  10,860  volts  is  the  generator  voltage. 


hence, 
When 


ii  =  0,  e  =  10,000,  eQ  =  10,860,  let  i2  =  i: 
these  values  substituted  give 

117.81  X  106  =  (10,000  +  20  i)2  +  100  i2 

=  100  X  106  +  400  i  X  103  +  500  *2, 
or, 

i  =  44.525  -  1.25  i2  10~3; 

this  equation  is  best  solved  by  approximation,  and  then  gives 

p  =  42.3  amp.  reactive  lagging  current  at  no  load. 
Since 

eo2  =  (e  +  rii  +  xi2)2  +  (riz  -  xii)2, 

it  follows  that 


e  =  Veo2  —  (™*2  —  xii)2  —  (rii  +  2^2); 


or, 


e  =  V117.81  +  106  -  (10  i*  -  20  n)2  -  (10  ii  +  20  t2). 
Substituting  herein  the  values  of  ii  and  i2  gives  e. 


t'l 

tz 

e 

0 

42.3 

10,000 

25 

28.2 

10,038 

50 

14.1 

10,038 

75 

0 

10,000 

100 

-14.1 

9,922 

125 

-28.2 

9,803 

80.  (2)  A  constant  voltage  eQ  is  impressed  upon  a  trans- 
mission line  of  impedance  Z  =  r  +  jx  =  10  +  10  j.  The  vol- 
tage at  the  receiving  end  shall  be  10,000  at  no  load  as  well  as  at 
full  load  of  100  amp.  power  current.  At  full  load  the  total 
current  shall  be  in  phase  with  the  e.m.f.  at  the  receiving  end, 
and  at  no  Load  a  lagging  current  of  50  amp.  is  permitted.  How 
much  additional  reactance  x0  is  to  be  inserted,  what  must  be  the 
generator  voltage  e0,  and  what  will  be  the  voltage  e  at  the  receiv- 


96       ELEMENTS  OF  ELECTRICAL  ENGINEERING 


ing  end  at  %  load  and  at  1J^  load,  if  the  reactive  current  varies 
proportionally  with  the  load? 

Let  XQ  =  additional  reactance  inserted  in  circuit. 
Let  I  —  i\  —  jiz  =  current. 
Then 
e02  =  (e  +  rii  +  x^)2  +  (na  -  Xiii)  2=  (e  +  10  ii  +  x^Y 

+  (10  i2  - 
where 

Xi  =  x  +  #o  =  total  reactance  of  circuit  between  e  and  eQ. 
At  no  load, 

ii  =  0,  i2  =  50,  e  =  10,000; 
thus,  substituting, 

e02  =  (10,000  +  50  zi)2  +  250,000. 
At  full  load, 

ii  =  100,  ia  =  0,  e  =  10,000; 
thus,  substituting, 

eo2  =  121  X  106  +  10,000  xj. 
Combining  these  gives 

(10,000  +  50  zi)2  +  250,000  =  121  X  106  +  10,000  a^i2; 
hence, 

xi  =  66.5  +  40.8 

=  107.3  or  25.7; 
thus 

XQ  =  Xi  —  x  =  97.3  or  15.7  ohms  additional  reactance. 

Substituting 
xi  =  25.7 
gives 

e02  =  (e  +  10  ^  +  25.7  i2)2  +  (10  i*  -  25.7  ii)2, 

but  at  full  load 

ii  =  100,  i2  =  0,  e  =  10,000, 
which  values  substituted  give 

e02  =  121  X  106  +  6.605  X  106  =  127.605  X  106, 

eo  =  11,300,  generator  voltage. 
Since 

e  =  vV  -  (10  12  -  25.7  *i)2  -  (10  ii  +  25.7  i2), 
it  follows  that 


6  =  V127.605  X  106  -  (10  it  -  25.7  ii)2  -  (10  ii  +  25.7  t2). 
Substituting  for  ii  and  z*2  gives  e. 


PHASE  CONTROL  OF  TRANSMISSION  LINES     97 


u 

ti 

e 

0 

50 

10,000 

50 

25 

10,105 

100 

0 

10,XKX) 

150 

-25 

9,658 

81.  (3)  In  a  circuit  whose  voltage  e0  fluctuates  by  20  per 
cent,  between  1800  and  2200  volts,  a  synchronous  motor  of 
internal  impedance  Z0  =  r0  +  jx0  =  0.5  +  5  j  is  connected 
through  a  reactive  coil  of  impedance  Z\  =  r\  +  jx\  =  0.5  -f-  10  j 
and  run  light,  as  compensator  (that  is,  generator  of  reactive 
currents).  How  will  the  voltage  at  the  synchronous  motor 
terminals  e\,  at  constant  excitation,  that  is,  constant  counter 
e.m.f.  e  =  2000,  vary  as  function  of  e$  at  no  load  and  at  a  load 
of  i  =  100  amp.  power  current,  and  what  will  be  the  reactive 
current  in  the  synchronous  motor? 

Let  I  =  ii  —  jiz  =  current  in  receiving  circuit  of  voltage  e\. 

Of  this  current  1,—jiz  is  taken  by  the  synchronous  motor  of 
counter  e.m.f.  'e,  and  thus 
EI  =  e  —  Zoji2 

=  e  +  X0i2  -  jr0i2'} 
or,  reduced, 

e^=  (e  +  xoit)2  +  rjif. 
In  the  supply  circuit  the  voltage  is 
Eo  =  Ei  +  IZl 

=  e  +  xoi*  -  jrQi2  +  (ii-jiz)  (TI  +  jxi) 

=  [e  +  riii  +  (xQ  +  xi)  iz]  —  j  [(rQ  +  TI)  i2  —      ' 

or,  reduced, 

eo2  =  [e  +  riti  +  (XQ  +  xi)  itf  +  [(r0  +  fi)  i2 


Substituting  in  the  equations  for  e^  and  e02  the  above  values 
of  r0  and  XQ:  at  no  load,  i\  =  0,  we  have 


6l2  =  (e  +  5  z2)2  +  0.25  i22  and  e02  =  (e  + 
at  full  load,  i\  =  100,  we  have 

ei2  =  (e  +  5  i2)2  +  0.25  z22, 


+ 


=  (e  +  50  +  15 


-  1000)2, 


98       ELEMENTS  OF  ELECTRICAL  ENGINEERING 


and  at  no  load,  i\  =  0,  substituting  e  =  2000,  we  have 

d2  =  (2000  +  5  i2)2  +  0.25  z22, 
eo2  =  (2000  +  15  *2)2  +  *22; 

at  full  load,  i\  =  100,  we  have 

ea2  =  (2000  +  5  *2)2  +  0.25  *22, 

e02  =  (2050  +  15  z2)2  +  fe  -  1000)2. 

Substituting  herein  e0  =  successively  1800,  1900,  2000,  2100, 
2200,  gives  values  of  i'2,  which,  substituted  in  the  equation  for 
ei2,  give  the  corresponding  values  of  ei  as  recorded  in  the  follow- 
ing table. 

As  seen,  in  the  local  circuit  controlled  by  the  synchronous 
compensator,  and  separated  by  reactance  from  the  main  circuit 
of  fluctuating  voltage,  the  fluctuations  of  voltage  appear  in  a 
greatly  reduced  magnitude  only,  and  could  be  entirely  eliminated 
by  varying  the  excitation  of  the  synchronous  compensator. 


e  =  2000 

No  load               ii  =  0 

Full  load       '     11  -  100 

H 

ei 

H 

e\ 

1,800 

-13.3 

1,937 

-39 

1,810 

1,900 

-  6.7 

1,965 

-30.1 

1,850 

2,000 

0 

2,000 

-22 

1,885 

2,100 

+  6.7 

2,035 

-13.5 

1,935 

2,200 

+  13.3 

2,074 

-  6.5 

1,970 

17.  IMPEDANCE  AND  ADMITTANCE 

82.  In   direct-current    circuits    the    most    important    law   is 
Ohm's  law, 


e 

-i   or  e 

r 


ir,     or  r  =  -.> 


where  e  is  the  e.m.f.  impressed  upon  resistance  r  to  produce 
current  i  therein. 

Since  in  alternating-current  circuits  a  current  i  through  a 

resistance  r  may  produce  additional  e.m.fs.  therein,  when  apply- 

a 
ing  Ohm's  law,  i  —  -  to   alternating-current   circuits,  e  is  the 


IMPEDANCE  AND  ADMITTANCE  '  99 

total  e.m.f.  resulting  from  the  impressed  e.m.f.  and  all  e.m.fs. 
produced  by  the  current  i  in  the  circuit. 

Such  counter  e.m.fs.  may  be  due  to  inductance,  as  self-induc- 
tance, or  mutual  inductance,  to  capacity,  chemical  polarization, 
etc. 

The  counter  e.m.f.  of  self-induction,  or  e.m.f.  generated  by  the 
magnetic  field  produced  by  the  alternating  current  i,  is  repre- 
sented by  a  quantity  of  the  same  dimensions  as  resistance,  and 
measured  in  ohms:  reactance  x.  The  e.m.f.  consumed  by 
reactance  x  is  in  quadrature  with  the  current,  that  consumed 
by  resistance  r  in  phase  with  the  current. 

Reactance  and  resistance  combined  give  the  impedance, 


+  x2; 

or,  in  symbolic  or  vector  representation, 
Z  =  r  +  jx. 

In  general  in  an  alternating-current  circuit  of  current  i,  the 
e.m.f.  e  can  be  resolved  in  two  components,  a  power  component 
ei  in  phase  with  the  current,  and  a  wattless  or  reactive  com- 
ponent e2  in  quadrature  with  the  current. 

The  quantity 

e_i  _  power  e.m.f.,  or  e.m.f.  in  phase  with  the  current  _ 
i  current 

is  called  the  effective  resistance. 
The  quantity 

62  _  reactive  e.m.f.,  or  e.m.f.  in  quadrature  with  the  current  _ 
i  current 

is  called  the  effective  reactance  of  the  circuit. 
And  the  quantity 


21  =  Vr!2  +  x2 
or,  in  symbolic  representation, 

Zi  =  ri  +  jxi 

is  the  impedance  of  the  circuit. 

If  power  is  consumed  in  the  circuit  only  by  the  ohmic  resist- 
ance r,  and  counter  e.m.f.  produced  only  by  self-inductance,  the 
effective  resistance  TI  is  the  true  or  ohmic  resistance  r,  and  the 
effective  reactance  Xi  is  the  true  or  inductive  reactance  x. 


100     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

By  means  of  the  terms  effective  resistance,  effective  reactance, 
and  impedance,  Ohm's  law  can  be  expressed  in  alternating- 
current  circuits  in  the  form 

•  =  -  e  m 

y  /      9     T    ~    9;  ^    ' 

Zi      vVi2  +  Xi2 
or,  e  =  izi  =  i  V^i2  +  Zi2;  (2) 


or,  «!  =  Vri8  +  a;ia  =  p  (3) 

or,  in  symbolic  or  vector  representation, 


or,  E  =  IZl  =  /(n+jxi);  (5) 

7^7 

or,  Zi  =  ri  +  jzi  =  j-  (6) 

In  this  latter  form  Ohm's  law  expresses  not  only  the  intensity 
but  also  the  phase  relation  of  the  quantities;  thus 

ei  =  iri  =  power  component  of  e.m.f., 
ez  =  ix\  =  reactive  component  of  e.m.f. 

p 

83.  Instead  of  the  term  impedance  z  —  -  with  its  components, 

I? 

the  resistance  and  reactance,  its  reciprocal  can  be  introduced. 


e   "   z  ' 


which  is  called  the  admittance. 

The  components  of  the  admittance  are  called  the  conduc- 
tance and  the  susceptance. 

Resolving  the  current  i  into  a  power  component  i\  in  phase 
with  the  e.m.f.  and  a  wattless  component  iz  in  quadrature  with 
the  e.m.f.,  the  quantity 

i\_   _  power  current,  or  current  in  phase  with  e.m.f. 

e  e.m.f.  .  =  9 

is  called  the  conductance. 
The  quantity 

_*2_   _  reactive  current,  or  current  in  quadrature  with  e.m.f. 
e  e.m.f. 

is  called  the  susceptance  of  the  circuit. 

The  conductance  represents  the  current  in  phase  with  the 


IMPEDANCE  AND  ADMITTANCE  101 

e.m.f.,  or  power  current,  the  susceptance  the  current  in  quad- 
rature with  the  e.m.f.,  or  reactive  current. 

Conductance  g  and  susceptance  b  combined  give  the  admittance 

y  =  Vg2  +  62;  (7) 

or,  in  symbolic  or  vector  representation, 

Y  =  g  -  jb.  (8) 

Thus  Ohm's  law  can  also  be  written  in  the  form 

i  =  ey  =  e  Vg2  +  &2;  (9) 

or, 

i 


or, 

y  =   Vg*  +  V  =  7;  (11) 

or,  in  symbolic  or  vector  representation, 

I  =  EY  =  E(g-jb);  (12) 

or, 

E  =     - 


or, 

Y  =  g  -  jb  =  |-  (14) 

and  i\  =  eg  =  power  component  of  current, 

ii  =  eb  =  reactive  component  of  current. 

84.  According  to  circumstances,  sometimes  the  use  of  the 
terms  impedance,  resistance,  reactance,  sometimes  the  use  of  the 
terms  admittance;  conductance,  susceptance,  is  more  convenient. 

Since,  in  a  number  of  series-connected  circuits,  the  total 
e.m.f.,  in  symbolic  representation,  is  the  sum  of  the  individual 
e.m.fs.,  it  follows  that  in  a  number  of  series-connected  circuits 
the  total  impedance,  in  symbolic  expression,  is  the  sum  of  the 
impedances  of  the  individual  circuits  connected  in  series. 

Since,  in  a  number  of  parallel-connected  circuits,  the  total 
current,  in  symbolic  representation,  is  the  sum  of  the  individual 
currents,  it  follows  that  in  a  number  of  parallel-connected  cir- 
cuits the  total  admittance,  in  symbolic  expression,  is  the  sum 
of  the  admittances  of  the  individual  circuits  connected  in  parallel. 


102     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Thus  in  series  connection  the  use  of  the  term  impedance,  in 
parallel  connection  the  use  of  the  term  admittance,  is  generally 
more  convenient. 

Since  in  symbolic  representation 

Y  =  ^  (15) 

or,  ZY  =  1;  (16) 

that  is,  (r+jx)(g  -  jb)  =  1;  (17) 

it  follows  that 

(rg  +  xb)  -  j  (rb  -  xg)  =  1; 
that  is  rg  -f  zb  =  1, 

rb  -  xg  =  0. 

r  =  -f±-  =  ±,  (18) 


6  =  JTqirji  =  •#>  (21) 

or,  in  absolute  values, 

y  =    '  (22) 


(23) 

(r2  +  x2)(^2  +  62)  =  1.  (24) 

Thereby  the  admittance  with  its  components,  the  conduc- 
tance and  susceptance,  can  be  calculated  from  the  impedance 
and  its  components,  the  resistance  and  reactance,  and  inversely. 

If  x  =  0,  z  =  r  and  g  =  —  ,  that  is,  g  is  the  reciprocal  of  the 

resistance  in  a  non-inductive  circuit;  not  so,   however,  in  an 
inductive  circuit. 

EXAMPLES 

85.  (1)  In  a  quarter-phase  induction  motor  having  an  im- 
pressed e.m.f.  e  =  110  volts  per  phase,  the  current  is  /0  =  ii  — 
jiz  =  100  —  100  j  at  standstill,  the  torque  =  D0. 

The  two  phases  are  connected  in  series  in  a  single-phase  cir- 
cuit of  e.m.f.  e  =  220,  and  one  phase  shunted  by  a  condenser  of 
1  ohm  capacity  reactance. 

What  is  the  starting  torque  D  of  the  motor  under  these  con- 
ditions, compared  with  Z>0,  the  torque  on  a  quarter-phase  cir- 


IMPEDANCE  AND  ADMITTANCE  103 

cuit,  and  what  the  relative  torque  per  volt-ampere  input,  if  the 
torque  is  proportional  to  the  product  of  the  e.m.fs.  impressed 
upon  the  two  circuits  and  the  sine  of  the  angle  of  phase  dis- 
placement between  them? 

In  the  quarter-phase  motor  the  torque  is 

D0  =  ae2  =  12,100  a, 
where  a  is  a  constant.     The  volt-ampere  input  is 


Qo  =  2  e  Vii2  +  i22  =  31,200; 

hence,   the  "apparent  torque  efficiency,"  or  torque  per  volt- 
ampere  input, 

rjQ  =  D*  =  0.388  a. 


The  admittance  per  motor  circuit  is 
the  impedance  is 


Y  =      =  0.91  -  0.91  j, 

e 


_  110  (100  +  100  j)         _055+055/ 

I  ~  100-  100  j  ~  (100.-100j)(100+100j)~ 

the  admittance  of  the  condenser  is 

Yo  =  j; 

thus,  the  joint  admittance  of  the  circuit  shunted  by  the  con- 
denser is 

Yi=  Y  +  7o  =  0.91  -  0.91  j  +  j 

=  0.91  +0.09  j; 
its  impedance  is 

7       J_       _  L_  0.91-  0.09  j 

Zl  ~  F,  ~  0.91  +  0.09  j  ~  0.9P  +  0.092  =  X  3> 

and  the  total  impedance  of  the  two  circuits  in  series  is 

Z2  =  Z  +  Zl 

=  0.55  +  0.55  j  +  1.09  -  0.11  j 
=  1.  64  +  0.44  j. 

Hence,  the  current,  at  impressed  e.m.f.  e  =  220. 

r      .        ..         e  220  220  (1.64-  0.44  j) 

!ti      ^2  ~  Z2~  1.64  +  0.44  j~       1.642  + 
=  125  -  33.5  j; 


104     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

or,  reduced, 

/  =  V1252  +  33.52 
=  129.4  amp. 

Thus,  the  volt-ampere  input, 

Q  =  el  =  220  X  129.4 
=  28,470. 

The  e.m.fs.  acting  upon  the  two  motor  circuits  respectively  are 
Ei  =  /Zi  =  (125  -  33.5  j)  (1.09  -  0.11  j)  =  132.8  -  50.4  j 

and 

E'  =  IZ  =  (125  -  33.5  j)  (0.55  +  0.55  j)  =  87.2  +  50.4  j. 

Thus,  the  tangents  of  their  phase  angles  are 

50  4 
tan  0i  =  +        -    =  +  0.30;  hence,  Bl  =  +  21°; 


50  4 
tan  tf  =  -  =  -  0.579;  hence,  6'  =  -  30°; 


and  the  phase  difference, 

0  =  0i 
The  absolute  values  of  these  e.m.fs.  are 


'  =  51°. 


ei  =  x/132.8  +  50.42  =  141.5 
and 

e'  =  V87.22  -  50.42  =  100.7; 

thus,  the  torque  is 

D  =  ae\e'  sin  6 
=  11, 100  a; 

and  the  apparent  torque  efficiency  is 

_D  11,100  a 
"  ~  Q  WTO" 

Hence,   comparing  this   with   the   quarter-phase    motor,    the 
relative  torque  is 

D_  =  11,100  a 
Do      12,100  a 

and  the  relative  torque  per  volt-ampere,  or  relative  apparent 
torque  efficiency,  is 

it       0.39  a 


0.388  a 


=  1.005. 


IMPEDANCE  AND  ADMITTANCE  105 

86.  (2)  At  constant  field  excitation,  corresponding  to  a 
nominal  generated  e.m.f.  JSQ  —  12,000,  a  generator  of  synchro- 
nous impedance  ZQ  =  r0  +  J^o  =  0.6  +  60  j  feeds  over  a  trans- 
mission line  of  impedance  Z\  =  ri  +  jx\  =  12  '+  18  j,  and  of 
capacity  susceptance  0.003,  a  non-inductive  receiving  circuit. 
How  will  the  voltage  at  the  receiving  end,  e,  and  the  voltage  at 
the  generator  terminals,  e\,  vary  with  the  load  if  the  line  capacity 
is  represented  by  a  condenser  shunted  across  the  middle  of  the 
line? 

Let  I  =  i  =  current  in  receiving  circuit,  in  phase  with  the 
e.m.f.,  E  =  e. 

The  voltage  in  the  middle  of  the  line  is 


=  e  +  6  i  +  9  ij. 

The  capacity  susceptance  of  the  line  is,  in  symbolic  expression, 
Y  =  0.003  j;  thus  the  charging  current  is 

72  =  E2Y  =  0.003  j  (e  +  6  i  +  9  ij) 

=  0.027  i  +  j  (0.003  e  +  0.018  i), 

and  the  total  current  is 

/!  =  I  +  72  =  0.973  i  +  j  (0.003  e  +  0.018  i). 
Thus,  the  voltage  at  the  generator  end  of  the  line  is 


=  e  +  6  i  +  9  ij  +  (6  +  9  j)[0.973  i  +  j  (0.003  e  + 

0.018  i)] 
=  (0.973  *  +  11.68  i)  +  j  (17.87  i  +  0.018  e), 

and  the  nominal  generated  e.m.f.  of  the  generator  is 

E0  =  E!  +  Zo|i 
=  (0.973  e  +  11.68  i)  +  j  (17.87  i  +  0.018  e)  +  (0.6  +  60  j) 

[0.973  t  +  j  (0.003  e  +  0.018  t)] 
=  (0.793  e  +  11.18  i)  +  j  (76.26  i  +  0.02  e); 

or,  reduced,  and  e0  =  12,000  substituted, 

e02    =  144  x  106  =  (0.793  e  +  11.18  i)2  +  (76.26  i  +  0.02  e)2; 
thus, 

e2  +  33  ei  +  9450  i2  =  229  X  106, 

e  =  -  16.5  i  +  V229  X  106  -  9178  1*, 


106     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

and 

ei  =  V(0.973e  +  11.68  *)2  -f  (17.87 i  +  O.OlSe)2; 
at 

i  =  0,  e  =  15,133,  a  =  14,700; 
at 

e  =  0,  i  =  155.6,  d  =  3327. 


P  )WER  CURRENT  REC'D  AMP 


V 


*OLT8 

5000 


3000 


5000 


3000 


2000 


10   20   JO   40   50   DO   70   80   90   100  110  120  130  140  150 

FIG.  40. — Reactive  load  characteristics  of  a  transmission  line  fed   by 
synchronous  generator  with  constant  field  excitation. 

Substituting  different  values  for  i  gives 


i 

' 

ei 

i 

e 

ei 

0 

15,133 

14,700 

100 

10,050 

11,100 

25 

14,488 

14,400 

125 

7,188 

8,800 

50 

13,525 

13,800 

150 

2,325 

4,840 

75 

12,063 

12,730 

155.6 

0 

3,327 

which  values  are  plotted  in  Fig.  40. 

18.  EQUIVALENT  SINE  WAVES 

87.  In  the  preceding  chapters,  alternating  waves  have  been 
assumed  and  considered  as  sine  waves. 


EQUIVALENT  SINE  WAVES  107 

The  general  alternating  wave  is,  however,  never  completely, 
frequently  not  even  approximately,  a  sine  wave. 

A  sine  wave  having  the  same  effective  value,  that  is,  the 
same  square  root  of  mean  squares  of  instantaneous  values,  as  a 
general  alternating  wave,  is  called  its  corresponding  "equivalent 
sine  wave."  It  represents  the  same  effect  as  the  general  wave. 

With  two  alternating  waves  of  different  shapes,  the  phase 
relation  or  angle  of  lag  is  indefinite.  Their  equivalent  sine 
waves,  however,  have  a  definite  phase  relation,  that  which 
gives  the  same  effect  as  the  general  wave,  that  is,  the  same 
mean  (ei). 

Hence  if  e  =  e.m.f.  and  i  =  current  of  a  general  alternating 
wave,  their  equivalent  sine  waves  are  defined  by 


e0  =  -\Anean  (e2), 

io  =  A/mean  (i2); 
and  the  power  is 

Po  =  eQiQ  cos  eoiQ  =  mean  (ei)', 
thus, 

mean  (ei) 

COS  €QIQ   =   —  /  - 

Vmean  (e2)  v  mean  (i2) 

Since  by  definition  the  equivalent  sine  waves  of  the  general 
alternating  waves  have  the  same  effective  value  or  intensity 
and  the  same  power  or  effect,  it  follows  that  in  regard  to  inten- 
sity and  effect  the  general  alternating  waves  can  be  represented 
by  their  equivalent  sine  waves. 

Considering  in  the  preceding  the  alternating  currents  as  equiva- 
lent sine  waves  representing  general  alternating  waves,  the 
investigation  becomes  applicable  to  any  alternating  circuit 
irrespective  of  the  wave  shape. 

The  use  of  the  terms  reactance,  impedance,  etc.,  implies  that 
a  wave  is  a  sine  wave  or  represented  by  an  equivalent  sine 
wave. 

Practically  all  measuring  instruments  of  alternating  waves 
(with  exception  of  instantaneous  methods)  as  ammsters,  volt- 
meters, wattmeters,  etc.,  give  not  general  alternating  waves 
but  their  corresponding  equivalent  sine  waves. 

EXAMPLES 

88.  In  a  25-cycle  alternating-current  transformer,  at  1000 
volts  primary  impressed  e.m.f.,  of  a  wave  shape  as  shown  in 


108     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


e 

§M 


»OCOOI>.C^O5(NCOOOOi'—  l 

i—  1  CO  CO  CO  »H    i—  1  <M  CO  IO  !>•  <J5  i—  1  r-tl>  OQ 


'^  CO  CO  C^J  ^H 


>O        CO        iQ  CO  C^  O  O5  CO  CO  iQ  CO  O  <N  00  O 
OiOO^Otoc^O 


S     4 


Ob-?—  ii>TfitOOOiOiOO 


^          rH  00  00  T-( 

c  co"  co"co"co'Nco>i-r 


^^^ooooooo 


1— 


«O  ..»  00 

00^2   ^ 
^  ^  se  ^ 


oo  oo  . 

<  r-l  CO  ?CI> 

i-H   QJ  CO 


EQUIVALENT  SINE  WAVES 


109 


Fig.  41  and  Table  I,  the  number  of  primary  turns  is  500,  the 
length  of  the  magnetic  circuit  50  cm.,  and  its  section  shall  be 
chosen  so  as  to  give  a  maximum  density  B  =  15,000. 

At  this  density  the  hysteretic  cycle  is  as  shown  in  Fig.  42  and 
Table  II. 


FIG.  41. — Wave-shape  of  e.m.f.  in  example  88. 

What  is  the  shape  of  current  wave,  and  what  the  equivalent 
sine  waves  of  e.m.f.,  magnetism,  and  current? 
The  calculation  is  carried  out  in  attached  table. 

TABLE  II 


/ 

B 

0                        ±8 

,000 

2 

+  10,400 

-    2,500 

4 

+  11,700 

+    5,800 

6 

+  12,400 

+    9,300 

8 

+  13,000 

+  11,200 

10 

+  13,500 

+  12,400 

12 

+  13,900 

+  13,200 

14                +  14,200 

+  13,800 

16                +  14,500 

+  14,300 

18 

+  14,800 

+  14,700 

20 

+  15,000 

In  column  (1)  are  given  the  degrees,  in  column  (2)  the  relative 
values  of  instantaneous  e.m.fs.,  e  corresponding  thereto,  as  taken 
from  Fig.  41. 

Column   (3)   gives  the  squares  of  e.     Their  sum  is  24,939; 

24  939 
thus  the  mean  square,    .  '  *  -  =  1385.5,  and  the  effective  value, 

lo 


110     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Since  the  effective  value  of  impressed  e.m.f.  is  =  1000,  the 

1 000 
instantaneous  values  are  eQ  =  e^-^  as  given  in  column  (4). 

Since  the  e.m.f.  e0  is  proportional  to  the  rate  of  change  of 
magnetic  flux,  that  is,  to  the  differential  coefficient  of  B}  B  is 
proportional  to  the  integral  of  the  e.m.f.,  that  is,  to  Se0  plus 
an  integration  constant.  2e0  is  given  in  column  (5),  and  the 
integration  constant  follows  from  the  condition  that  B  at  180° 


FIG.  42. — Hysteretic  cycle  in  example  88. 

must  be  equal,  but  opposite  in  sign,  to  B  at  0°.     The  integration 
constant  is,  therefore, 

1  SO  -it      n    i  ^ 


and  by  subtracting  7324  from  the  values  in  column  (5)  the 
values  of  B'  of  column  (6)  are  found  as  the  relative  instantaneous 
values  of  magnetic  flux  density. 

Since  the  maximum  magnetic  flux  density  is  15,000  the  in- 

15  000 
stantaneous  values  are  B  =  B'      '    .  ,  plotted  in  column  (7). 

From  the  hysteresis  cycle  in  Fig.  42  are  taken  the  values  of 
magnetizing  force  /,  corresponding  to  magnetic  flux  density  B. 
They  are  recorded  in  column  (8),  and  in  column  (9)  the  instan- 
taneous values  of  m.m.f.  F  =  If,  where  I  =  50  =  length  of 
magnetic  circuit. 


EQUIVALENT  SINE  WAVES 


111 


i  =  — ,   where  n  =  500  =  number  of    turns    of    the  electric 

circuit,  gives  thus  the  exciting  current  in  column  (10) . 

Column   (11)   gives  the  squares  of  the  exciting  current,  i2. 

25  85 
Their  sum  is  25.85;  thus  the  mean  square,      '       =  1.436,  and 

lo 

the   effective   value   of  exciting   current,   i'  =  Vl.436  =  1.198 
amp. 

Column  (12)  gives  the  instantaneous  values  of  power,  p  =  ieo. 


Their  sum  is  4766;  thus  the  mean  power,  p'  = 


4766 
18 


=  264.8. 


FIG.  43. — Waves  of  exciting  current. 
Power  and  flux  density  corresponding 
to  e.m.f .  in  Fig.  41  and  hysteretic  cycle 
in  Fig.  42. 


FIG.  44. — Corresponding  sine  waves 
for  e.m.f.  and  exciting  current  in  Fig. 
43. 


Since  p'  =  i'e'Q  cos  0, 

where  e'0  and  i'  are  the  equivalent  sine  waves  of  e.m.f.  and  of 
current  respectively,  and  0  their  phase  displacement,  substitut- 
ing these  numerical  values  of  p',  er,  and  i',  we  have 

264.8  =  1000  X  1.198  cos  6. 
hence, 

cos  0  =  0.221, 

6  =  77.2°, 


112     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

and  the  angle  of  hysteretic  advance  of  phase, 

a  =  90°  -  0  =  12.8°. 
The  hysteresis  current  is  then 

i'  cos  e  =  0.265, 
and  the  magnetizing  current, 

i'  sin  0  =  1.165. 

Adding  the  instantaneous  values  of  e.m.f.  eQ  in  column  (4) 

14  648 
gives  14,648;  thus  the  mean  value,  —  f-r—  =  813.8.     Since  the 

J-O 

effective  value  is  1000,  the  mean  value  of  a  sine  wave  would  be 

2  -v/2 
1000-      -  =  904;  hence  the  form  factor  is 

7T 

904 
7  = 


Adding  the  instantaneous  values  of  current  i  in  column  (10), 
irrespective  of  their  sign,  gives  17.17;  thus  the  mean  value, 

17.17 

'       =  0.954.     Since    the    effective    value  =  1.198,    the    form 

lo 

factor  is  1.198  2  V2 

7  =  0954  — 

The  instantaneous  values  of  e.m.f.  e0,  current  i,  flux  density 
B  and  power  p  are  plotted  in  Fig.  43,  their  corresponding  sine 
waves  in  Fig.  44. 

19.  FIELDS  OF  FORCE 

89.  When  an  electric  current  flows  through  a  conductor, 
power  is  consumed  and  heat  produced  inside  of  the  conductor. 
In  the  space  outside  and  surrounding  the  conductor,  a  change 
has  taken  place  also,  and  this  space  is  not  neutral  and  inert 
any  more,  but  if  we  try  to  move  a  solid  mass  of  metal  rapidly 
through  it,  the  motion  is  resisted,  and  heat  produced  in  the 
metal  by  induced  currents.  Materials  of  high  permeability, 
as  iron  filings,  brought  into  this  space  arrange  themselves 
in  chains;  a  magnetic  needle  is  moved  and  places  itself  in  a 
definite  direction.  Due  to  the  passage  of  the  current  in  the 
conductor,  there  are  therefore  in  the  spaces  outside  of  the  con- 
ductor —  where  the  current  does  not  flow  —  forces  exerted,  and 


FIELDS  OF  FORCE  113 

this  space  then  is  not  neutral  space,  but  has  become  a  field  of 
force,  and  the  cause  of  the  field,  in  this  case  the  electric  current 
in  the  conductor,  is  its  "motive  force."  As  in  this  case  the 
actions  exerted  in  the  field  of  force  are  magnetic,  the  space 
surrounding  a  conductor  traversed  by  a  current  is  a  field  of 
magnetic  force,  and  the  current  in  the  conductor  is  the  magneto- 
motive force. 

In  the  space  surrounding  a  ponderable  mass,  as  our  earth, 
forces  are  exerted  on  other  masses — which  cause  the  stone  to 
fall  toward  the  earth,  and  water  to  run  down  hill — and  this 
space  thus  is  a  field  of  gravitational  force,  the  earth  the  gram- 
motive  force. 

In  the  space  surrounding  conductors  having  a  high  potential 
difference,  we  observe  a  field  of  dielectric  force,  that  is,  electro- 
static or  dielectric  forces  are  exerted,  and  the  potential  difference 
between  the  conductors  is  the  electromotive  force  of  the  dielectric 
field. 

The  force  exerted  by  the  earth  as  gravimotive  force,  on  any 
mass  in  the  gravitational  field  of  the  earth,  causes  the  mass  to 
move  with  increasing  rapidity.  The  direction  of  motion  then 
shows  the  direction  in  which  the  force  acts,  that  is,  the  direc- 
tion of  the  gravitational  field.  The  force  g,  which  the  field  exerts 
on  unit  mass,  that  is,  the  acceleration  of  the  mass,  measures 
the  intensity  of  the  field:  in  the  gravitational  field  of  the  earth 
981  cm  g  sec.  The  force  acting  upon  a  mass  m,  then,  is:F  = 
gm,  and  is  called  the  weight  of  the  mass. 

In  the  same  manner,  in  the  magnetic  field  of  a  current  as 
magnetomotive  force,  the  intensity  H  of  the  magnetic  field  is 
measured  by  the  force  F  which  the  field  exerts  on  a  magnetic 
mass  or  pole  strength  m:  F  =  Hm;  the  intensity  K  of  the  di- 
electric field  of  a  potential  difference  as  electromotive  force  is 
measured  by  the  force  F  exerted  upon  an  electric  pole  strength 
e:  F  =  Ke',  the  direction  of  the  force  represents  the  direction  of 
the  field  of  force. 

90.  This  conception  of  the  field  of  force  is  one  of  the  most 
important  and  fundamental  ones  of  all  sciences  and  applied 
sciences:  a  condition  of  space,  brought  about  by  some  exciting 
cause  or  motive  force,  whereby  the  space  is  not  neutral  any 
more,  but  capable  of  exerting  forces  on  anything  susceptible  to 
these  forces:  mechanical  forces  on  masses  in  a  gravitational  field, 
magnetic  forces  on  magnetic  materials  in  a  magnetic  field, 


114     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


A. — A  photograph  of  a  mica-filing  map  of  the  dielectric  lines  of  force- 
between  two  cylinders. 


B. — A  photograph  of  an  iron-filing  map  of  the  magnetic  lines  of  force  about. 

two  cylinders. 


C. — A  photographic  superposition  of  A  and  B  representing  the  magnetic- 
and  dielectric  fields  of  the  space  surrounding  two  conductors  which  are; 
carrying  energy. 

FIG.  45. 


FIELDS  OF  FORCE 


115 


dielectric  forces  on  dielectrics  in  a  dielectric  field,  etc.  The  field 
of  force  then  is  characterized  by  having,  at  any  point,  a  definite 
direction — the  direction  in  which  the  force  acts — and  a  definite 
intensity,  to  which  the  forces  are  proportional. 

Such  fields  of  force  can  be  graphically  represented  by  lines 
showing  the  direction  in  which  the  force  acts:  the  lines  of  force 
and,  at  right  angles  thereto,  the  equipotential  lines  or  surfaces, 
as  the  direction  in  which  no  force  acts.  Thus  the  lines  of  gravita- 
tional force  of  the  earth  are  the  verticals,  the  equipotential  sur- 
faces, or  level  surfaces,  are  the  horizontals.  Such  pictures  of  a 
field  of  force  also  illustrate  the  intensity:  where  the  lines  of  force 
and  therefore  the  equipotential  lines  come  closer  together,  the 
field  is  more  intense,  that  is,  the  forces  greater. 


FIG.  46. — A  mathematical  plot  of  fields  shown  in  C. 

Magnetic  fields  may  be  demonstrated  by  iron  filings  brought 
into  the  field;  dielectric  fields  by  particles  of  a  material  of  high 
specific  capacity,  such  as  mica.  Fig.  45  shows  the  dielectric 
field  of  a  pair  of  parallel  conductors,  the  magnetic  field  between 
these  conductors,  and  their  combination.  Fig.  46  shows  the  same 
as  calculated. 

As  further  illustration,  Fig.  47  shows,  from  observation,  half 
of  the  dielectric  field  between  a  rod  with  circular  disc,  as  one 
terminal,  passing  symmetrically  through  the  center  of  a  cylinder 
placed  in  a  circular  hole  in  a  plate  as  other  terminal:  the 
lines  of  force  pass  from  terminal  to  terminal;  the  equipotential 
surfaces  intersect  at  right  angles  (A  10,292). 

91.  In  electrical  engineering  we  have  to  deal  with  the  electrical 
quantities:  voltage,  current,  resistance,  etc.;  the  magnetic  quan- 


116     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

titles:  magnetic  flux,  field  intensity,  permeability,  etc.;  and  the  di- 
electric quantities:  dielectric  flux,  field  intensity,  permittivity,  etc. 
The  electric  current  is  the  magnetomotive  force  F  which  produces 
the  magnetic  field,  acting  upon  space.  It  is  expressed  in  amperes, 
or  rather  in  ampere-turns,  and  thus  is  an  electrical  quantity,  its 


Rod 


V-0 


Plane 


Hole 


FIG.  47. — Observed  dielectric  field. 


unit  being  determined  by  the  unit  of  current,  as  the  ampere-turn 
equal  to  10"1  absolute  units. 

The  magnetomotive  force  per  unit  length  of  the  magnetic 
circuit  then  is  the  magnetizing  force  or  magnetic  gradient  f, 
in  ampere-turns  per  centimeter,  hence  still  an  electrical  quantity. 

Proportional   thereto,    and   of  the   same    dimension,    is    the 


FIELDS  OF  FORCE  117 

magnetic  field  intensity  H.  It  differs  from  the  magnetic  gradient 
merely  by  a  numerical  factor  4  TT;  H  =  4?r/  10"1.  Magnetic  field 
intensity  is  a  magnetic  quantity,  and  its  unit  defined  by  the 
magnetic  forces  exerted  in  the  field,  thus  different  from  the  unit 
of  magnetic  gradient,  which  is  determined  by  the  unit  of  electric 
current;  hence  the  factor  4  IT.  The  factor  10"1  merely  reduces 
from  amperes  to  absolute  unit. 

If  then  v  is  the  magnetic  conductivity  of  the  material  in  the 
magnetic  field,  called  its  permeability,  B  =  pH  is  the  magnetic 
flux  density,  and  the  total  magnetic  flux  <1>  is  given  by  the 
density  B  times  the  area  or  section  of  the  flux. 

Or,  passing  directly  from  the  magnetomotive  force  F  to  the 

F 

magnetic  flux,  by  the  conception  of  the  magnetic  circuit:  3>  =  „> 

where  R  is  the  magnetic  resistance,  or  reluctance  of  the  magnetic 
circuit. 

R  is  an  electric  quantity,  and  does  not  contain  the  4  TT. 

In  the  dielectric  field,  the  potential  difference  e  is  the  electro- 
motive force  expressed  in  volts.  The  electromotive  force  per 
unit  length  of  the  dielectric  circuit  is  the  electrifying  force  or 
voltage  gradient  or  dielectric  gradient  g,  expressed  in  volts  per 
centimeter.  This  is  still  an  electric  quantity. 

Proportional  thereto  by  a  numerical  factor  is  the  dielectric 

quantity:    dielectric  field  intensity   K  =    .      2,   and  if  k  is  the 

dielectric  conductivity  of  the  medium  in  the  dielectric  field, 
called  specific  capacity  or  permittivity,  the  dielectric  flux  density  is 
D  =  kK,  and  the  total  dielectric  flux  ^  is  flux  density  times  area. 

Here  again,  at  the  transition  from  the  electric  quantity 
"gradient"  to  the  dielectric  quantity  " field  intensity,"  a  numer- 
ical factor  4  irv2  enters,  the  one  quantity  being  based  on  the  volt 
as  unit,  the  other  on  unit  force  action,  v  is  the  velocity  of  light, 
3  X  1010,  and  the  factor  v2  the  result  of  the  convention  of  assum- 
ing the  permittivity  of  empty  space  as  unity. 

It  is  now  easy  to  remember,  where  in  the  electromagnetic 
system  of  units  the  factor  4-Tr  enters:  it  is  at  the  transition  from 
the  electrical  quantities  to  the  magnetic  or  dielectric  quantities, 
from  gradient  to  field  intensity. 

92.  The  dielectric  field  and  the  magnetic  field  are  analogous, 
and  to  magnetic  flux,  magnetic  field  intensity,  permeability,  as 
used  in  dealing  with  magnetic  circuits,  correspond  the  terms 


118     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

dielectric  flux,  dielectric  field  intensity,  permittivity,  as  used  in 
dealing  with  the  electrostatic  fields  of  high  potential  apparatus, 
as  transmission  insulators,  transformer  bushings,  etc.  The  fore- 
most difference  is  that  in  the  magnetic  field,  a  line  of  force  must 
always  return  into  itself  in  a  closed  circuit,  while  in  the  electro- 
static or  dielectric  field,  a  line  of  force  may  terminate  in  a  con- 
ductor. The  terminals  of  the  lines  of  electrostatic  flux,  ^  at  the 
conductor,  then  are  represented  by  the  conception  of  a  quantity 
of  electricity  or  electric  charge,  Q,  being  located  on  the  con- 
ductor. Thus,  at  the  terminal  of  the  line  of  unit  dielectric  flux, 
unit  electric  quantity  is  located  on  the  conductor. 

Dielectric  flux  ^  and  electric  quantity  or  charge  Q  thus  are 
identical,  and  merely  different  conceptions  of  the  dielectric 
circuit : 

Q  =  *. 

In  using  the  conception  of  electric  quantity  Q,  we  consider 
only  the  terminals  of  the  lines  of  dielectric  flux,  that  is,  deal 
merely  with  the  effect  of  the  dielectric  flux  on  the  electric  circuit 
which  produced  it.  This  conception  is  in  many  cases  more 
convenient,  but  it  necessarily  fails,  when  the  distribution  of  the 
dielectric  flux  in  the  dielectric  field  is  of  importance,  such  as  is  the 
case  when  dealing  with  high  dielectric  field  intensities,  approach- 
ing the  possibility  of  disruptive  effects  in  the  field  of  force,  or 
when  dealing  with  the  effect  produced  by  the  introduction  of  ma- 
terials of  different  permittivity  into  the  dielectric  field.  There- 
fore, with  the  increasing  importance  of  the  dielectric  field  in 
engineering,  the  conception  of  electric  quantity,  or  charge,  is 
gradually  being  replaced  by  the  conception  of  the  dielectric  flux 
and  the  dielectric  field,  analogous  to  the  magnetic  field,  which 
has  replaced  the  previous  conception  of  "  magnetic  poles." 

20.  NOMENCLATURE 

93.  The  following  nomenclature  and  symbols  of  the  quantities 
most  frequently  used  in  electrical  engineering  appears  most 
satisfactory,  and  is  therefore  recommended.  It  is  in  agreement 
with  the  Standardization  Rules  of  the  A.  I.  E.  E.,  but  as  far  as 
possible  standard  letters  have  been  used,  and  script  letters  avoided 
as  impracticable  or  at  least  inconvenient  in  writing  and  still 
more  in  typewriting.  Therefore  F  has  been  chosen  for  m.m.f., 
and  dielectric  field  intensity  changed  to  K.  Also,  a  few  symbols 
not  contained  in  the  Standardization  Rules  had  to  be  added. 


NOMENCLATURE 

TABLE  OP  SYMBOLS 


119 


Symbol 

Name 

Unit 

Character 

E,  e. 

Voltage 

Volt 

Electrical 

I,  i.  . 

Potential  difference 
Electromotive  force 
Current 

Ampere 

Electrical 

R,r 

Resistance 

Ohm 

Electrical 

x 

Reactance 

Ohm 

Electrical 

Z,z... 

Impedance 

Ohm 

Electrical 

a 

Conductance 

Mho 

Electrical 

b 

Susceptance 

Mho 

Electrical 

Y,y  
p  

Admittance 
Resistivity 

Mho 
Ohm-centimeter 

Electrical 
Electrical 

7  
$  

Conductivity 
Magnetic  flux 

Mho-centimeter 
Line;  kiloline;  megaline 

Electrical 
Magnetic 

£....... 
H 

Magnetic  density 
Magnetic    field    inten- 

Lines   per    cm.2;    kilo- 
lines  per  cm.2 
Lines  per  cm  2 

Magnetic 
Magnetic 

/*        •  • 

sity 
Permeability  (magnetic 

Magnetic 

/  

conductivity) 
Magnetic  gradient 

Ampere-turns  per  centi- 

Electrical 

F 

Magnetizing  force 
Magnetomotive  force 

meter. 
Ampere-turns 

Electrical 

R  

Reluctance     (magnetic 

Electrical 

L  
M  

S      ..     . 

resistance) 
Inductance 
Mutual  inductance 
Self-inductance 

Henry;  milhenry 
Henry;  milhenry 
Henry;  milhenry 

Magnetic 
Magnetic 
Magnetic 

*,Q  

D  
K  

Leakage  inductance 
Dielectric  flux 
Electric      quantity     or 
charge 
Dielectric  density 

Dielectric    field   inten- 

Lines of  dielectric  force 
Coulombs 

Dielectric  lines  per  cm.2 
Coulombs  per  cm.2 

Dielectric 

Dielectric 
Dielectric 

k 

sity 
Permittivity 

Dielectric 

Specific  capacity 

120     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

TABLE  OF  SYMBOLS.     Continued 


Symbol 

Name 

Unit 

Character 

9  

Dielectric  gradient 

Volts  per  centimeter 

Electrical 

Voltage  gradient 

Electrifying  force 

C  

Capacity 

Farad;  microfarad 

Dielectric 

P,P  

Power,  effect 

Watt;  kilowatt 

General 

W,w.... 

Energy,  work 

Joule;  kilo  joule 

General 

T,#  

Temperature 

Degrees  Centigrade 

General 

t  

Time 

Seconds 

General 

$,$,0... 

Time  angle 

Degrees  or  radians 

General 

<*,T  

Space  angle 

Degrees  or  radians 

General 

/  

Frequency 

Cycles  per  second 

General 

PART  II 

SPECIAL  APPARATUS 


INTRODUCTION 

1.  By  the  direction  of  the  energy  transmitted,  electric  machines 
have  been  divided  into  generators  and  motors.  By  the  character 
of  the  electric  power  they  have  been  distinguished  as  direct- 
current  and  as  alternating-current  apparatus. 

With  the  advance  of  electrical  engineering,  however,  these 
subdivisions  have  become  unsatisfactory  and  insufficient. 

The  division  into  generators  and  motors  is  not  based  on  any 
characteristic  feature  of  the  apparatus,  and  is  thus  not  rational. 
Practically  any  electric  generator  can  be  used  as  motor,  and 
conversely,  and  frequently  one  and  the  same  machine  is  used  for 
either  purpose.  Where  a  difference  is  made  in  the  construction, 
it  is  either  only  quantitative,  as,  for  instance,  in  synchronous 
motors  a  higher  armature  reaction  is  often  used  than  in  synchro- 
nous generators,  or  it  is  in  minor  features,  as  direct-current  motors 
usually  have  only  one  field  winding,  either  shunt  or  series,  while 
in  generators  frequently  a  compound  field  is  employed.  Further- 
more, apparatus  have  been  introduced  which  are  neither  motors 
nor  generators,  as  the  synchronous  machine  producing  wattless  lag- 
ging or  leading  current,  etc.,  and  the  different  types  of  converters. 

The  subdivision  into  direct-current  and  alternating-current 
apparatus  is  unsatisfactory,  since  it  includes  in  the  same  class 
apparatus  of  entirely  different  character,  as  the  induction  motor 
and  the  alternating-current  generator,  or  the  constant-potential 
commutating  machine  and  the  rectifying  arc  light  machine. 

Thus  the  following  classification,  based  on  the  characteristic 
features  of  the  apparatus,  as  adopted  by  the  A.  I.  E.  E.  Standard- 
izing Committee,  is  used  in  the  following  discussion.  It  refers 
only  to  the  apparatus  transforming  between  electric  and  electric 
and  between  electric  and  mechanical  power. 

1st.  Commutating  machines,  consisting  of  a  magnetic  field  and 
a  closed-coil  armature,  connected  with  a  multi-segmental 
commutator. 

121 


122     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

2d.  Synchronous  machines,  consisting  of  a  undirectional  mag- 
netic field  and  an  armature  revolving  relatively  to  the  mag- 
netic field  at  a  velocity  synchronous  with  the  frequency  of  the 
alternating-current  circuit  connected  thereto. 
.  3d.  Rectifying  apparatus,  that  is,  apparatus  reversing  the  direc- 
tion of  an  alternating  current  synchronously  with  the  frequency. 

4th.  Induction  machines,  consisting  of  an  alternating  mag- 
netic circuit  or  circuits  interlinked  with  two  electric  circuits  or 
sets  of  circuits  moving  with  regard  to  each  other. 

5th.  Stationary  induction  apparatus,  consisting  of  a  magnetic 
circuit  interlinked  with  one  or  more  electric  circuits. 

6th.  Electrostatic  and  electrolytic  apparatus  as  condensers  and 
polarization  cells. 

Apparatus  changing  from  one  to  a  different  form  of  electric 
energy  have  been  defined  as: 

A.  Transformers,  when  using  magnetism,  and  as 

B.  Converters,  when  using    mechanical  momentum  as  inter- 
mediary form  of  energy. 

The  transformers  as  a  rule  are  stationary,  the  converters 
rotary  apparatus.  Motor-generators  transforming  from  elec- 
trical over  mechanical  to  electric  power  by  two  separate  machines, 
and  dynamotors,  in  which  these  two  machines  are  combined  in 
the  same  structure,  are  not  included  under  converters. 

2.  (1)  Direct-current  commutating  machines  as  generators  are 
Usually  built  to  produce  constant  potential  for  railway,  incan- 
descent lighting,  and  general  distribution.  As  motors  commutat- 
ing machines  give  approximately  constant  speed — shunt  motors 
— or  large  starting  torque — series  motors. 

When  inserted  in  series  in  a  circuit,  and  controlled  so  as  to 
give  an  e.m.f.  varying  with  the  conditions  of  load  on  the  system, 
these  machines  are  "boosters,"  and  are  generators  when  raising 
the  voltage,  and  motors  when  lowering  it. 

Commutating  machines  may  be  used  as  direct-current  con- 
verters by  transforming  power  from  one  side  to  the  other  side  of 
a  three- wire  system. 

Alternating-current  commutating  machines  are  used  as  motors 
of  series  characteristic  for  railway  and  other  varying  speed  service, 
or  with  shunt  characteristic  for  constant  speed  and  adjustable 
speed  work,  especially  where  high  starting  torque  efficiency  is 
required.  They  usually  are  of  single-phase  type. 

(2)  While    in    commutating  machines  the   magnetic  field  is, 


INTRODUCTION  123 

almost  always  stationary  and  the  armature  rotating,  synchronous 
machines  were  built  with  stationary  field  and  revolving  armature, 
or  with  stationary  armature  and  revolving  field,  or  as  inductor 
machines  with  stationary  armature  and  stationary  field  winding 
but  revolving  magnetic  circuit.  Generally  now  the  revolving 
field  type  is  used. 

By  the  number  and  character  of  the  alternating  circuits  con- 
nected to  them  they  are  single-phase  or  polyphase  machines. 
As  generators  they  comprise  practically  all  single-phase  and  poly- 
phase alternating-current  generators;  as  motors  a  very  important 
class  of  apparatus,  the  synchronous  motors,  which  are  usually 
preferred  for  large  powers,  especially  where  frequent  starting 
and  considerable  starting  torque  are  not  needed.  Synchronous 
machines  may  be  used  as  compensators  or  synchronous  condensers, 
to  produce  wattless  current,  leading  by  over-excitation,  lagging 
by  under-excitation,  or  may  be  used  as  phase  converters  by  operat- 
ing a  polyphase  synchronous  motor  by  one  pair  of  terminals  from 
a  single-phase  circuit.  The  most  important  class  of  converters, 
however,  are  the  synchronous  commutating  machines,  to  which, 
therefore,  a  special  chapter  will  be  devoted  in  the  following. 

Inserted  in  series  to  another  synchronous  machine  or  synchro- 
nous converter,  and  rigidly  connected  thereto,  synchronous  ma- 
chines are  also  occasionally  used  as  boosters. 

Synchronous  commutating  machines  contain  a  unidirectional 
magnetic  field  and  a  closed  circuit  armature  connected  simul- 
taneously to  a  segmental  direct-current  commutator  and  by 
collector  rings  to  an  alternating  circuit,  generally  a  polyphase 
system.  Thus  these  machines  can  either  receive  alternating  and 
yield  direct-current  power  as  synchronous  converters  or  simply 
"  converters,"  or  receive  direct  and  yield  alternating-current 
power  as  inverted  converters,  or  driven  by  mechanical  power 
yield  alternating  and  direct  current  as  double-current  generators. 
Or  they  can  combine  motor  and  generator  action  with  their 
converter  action.  Thus  a  combination  is  a  synchronous  con- 
verter supplying  a  certain  amount  of  mechanical  power  as  a 
synchronous  motor.  Usually,  they  convert  from  three-phase  or 
single-phase  alternating  to  direct-current  power. 

(3)  Rectifying  machines  are  apparatus  which  by  a  synchro- 
nously revolving  rectifying  commutator  send  the  successive  half 
waves  of  an  alternating  single-phase  or  polyphase  circuit  in  the 
same  direction  into  the  receiving  circuit.  The  most  impor- 


124     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

tant  class  of  such  apparatus  were  the  open-coil  arc  light  ma- 
chines. They  have  been  practically  superseded  by  the  mercury 
arc  rectifier. 

(4)  Induction  machines   are   generally  used  as  motors,  poly- 
phase  or   single-phase.     In   this   case   they   run   at   practically 
constant  speed,  slowing  down  slightly  with  increasing  load.     As 
generators  the  frequency  of  the  e.m.f.  supplied  by  them  differs 
from  and  is  lower  than  the  frequency  of  rotation,  but  their  opera- 
tion depends  upon  the  phase  relation  of  the  external  circuit.     As 
phase  converters,  induction  machines  can  be  used  in  the  same 
manner  as  synchronous  machines.     Another  occasional  use  be- 
sides as  motors  is,  however,  as  frequency  converters,  by  changing 
from  an  impressed  primary  polyphase  system  to  a  secondary 
polyphase  system  of  different  frequency.     In  this  case,  when  low- 
ering the  frequency,  mechanical  energy  is  also  produced;  when 
raising  the  frequency,  mechanical  energy  is  consumed. 

(5)  The  most  important  stationary  induction  apparatus  is  the 
transformer,  consisting  of  two  electric  circuits  interlinked  with 
the  same  magnetic  circuit.     When  using  the  same  or  part  of  the 
same  electric  circuit  for  primary  and  secondary,  the  transformer 
is  called  an  auto-transformer  or  compensator.     When  inserted  in 
series  into  a  circuit,  and  arranged  to  vary  the  e.m.f.,  the  trans- 
former is  called  potential  regulator  or  booster.     The  variation  of 
secondary  e.m.f.  may  be  secured  by  varying  the  relative  number 
of  primary  and  secondary  turns,  or  by  varying  the  mutual  in- 
ductance between  primary  and  secondary  circuit,  either  elec- 
trically or  magnetically.     The  stationary  induction  apparatus 
with  one  electric  circuit  are  used  for  producing  wattless  lagging 
currents,  as  reactors,  reactive  or  choke  coils. 

(6)  Condensers  and  polarization  cells  produce  wattless  leading 
currents,  the  latter,  however,  usually  at  a  low  efficiency,  while 
the  efficiency  of  the   condenser  is  extremely  high,   frequently 
above  99  per  cent. ;  that  is,  the  loss  of  power  is  less  than  1  per  cent, 
of  the  apparent  volt-ampere  input. 

Unipolar,  or,  more  correctly,  non-polar  or  acyclic  machines 
are  apparatus  in  which  a  conductor  cuts  a  continuous  magnetic 
field  at  a  uniform  rate.  They  have  not  become  of  industrial 
importance. 

Regarding  apparatus  transforming  between  electric  energy 
and  forms  of  energy  differing  from  electric  or  mechanical  energy: 
The  transformation  between  electrical  and  chemical  energy  is 


INTRODUCTION  125 

represented  by  the  primary  and  secondary  battery  and  the  elec- 
trolytic cell;  the  transformation  between  electrical  and  heat 
energy  by  the  thermopile  and  the  electric  heater  or  electric  fur- 
nace; the  transformation  between  electrical  and  light  energy  by 
the  incandescent  and  arc  lamps. 

In  the  following  will  be  given  a  general  discussion  of  the  charac- 
teristics of  the  most  frequently  used  and  therefore  most  impor- 
tant classes  of  apparatus. 

A  further  discussion  and  calculation  of  these  apparatus  is  given 
in  "Theory  and  Calculation  of  Alternating  Current  Phenomena," 
while  a  discussion  of  those  characteristics  and  modifications  of 
these  apparatus,  which,  though  important,  are  less  frequently 
met,  and  a  discussion  of  the  numerous  less  common  types  of 
apparatus,  which  could  not  be  included  in  the  following,  is  given 
in  "Theory  and  Calculation  of  Electrical  Apparatus." 

Some  important  features,  as  the  nature  of  the  reactance  of 
apparatus,  mechanical  magnetic  forces,  wave  shape  distortions 
caused  by  some  features  of  design,  in  apparatus,  etc.,  are  dis- 
cussed in  "Theory  and  Calculation  of  Electric  Circuits." 


A.  SYNCHRONOUS  MACHINES 
I.  General 

3.  The  most  important  class  of  alternating-current  apparatus 
consists  of  the  synchronous  machines.  They  comprise  the 
alternating-current  generators,  single-phase  and  polyphase,  the 
synchronous  motors,  the  phase  compensators,  the  phase  con- 
verters, the  phase  balancers,  the  synchronous  boosters  and  the 
exciters  of  induction  generators,  that  is,  synchronous  machines 
producing  wattless  lagging  or  leading  currents,  and  the  con- 
verters. Since  the  latter  combine  features  of  the  commutating 
machines  with  those  of  the  synchronous  machines  they  will  be 
considered  separately. 

In  the  synchronous  machines  the  terminal  voltage  and  the 
generated  e.m.f.  are  in  synchronism  with,  that  is,  of  the  same 
frequency  as,  the  speed  of  rotation. 

These  machines  consist  of  an  armature,  in  which  e.m.f.  is 
generated  by  the  rotation  relatively  to  a  magnetic  field,  and  a 
continuous  magnetic  field,  excited  either  by  direct  current,  or  by 
the  reaction  of  displaced  phase  armature  currents,  or  by  per- 
manent magnetism. 

The  formula  for  the  e.m.f.  generated  in  synchronous  machines, 
commonly  called  alternators,  is 

E  =     S2irn3>  =  4. 


where  n  is  the  number  of  armature  turns  in  series  interlinked 
with  the  magnetic  flux  <1>  (in  megalines  per  pole),  /  the  frequency 
of  rotation  (in  hundreds  of  cycles  per  second),  E  the  e.m.f.  gen- 
erated in  the  armature  turns. 

This  formula  assumes  a  sine  wave  of  e.m.f.     If  the  e.m.f. 
wave  differs  from  sine  shape,  the  e.m.f.  is 

E  =  4.447/n<I>, 

2  -\/2 
where  y  =  form  factor  of  the  wave,  or  —     -  times  ratio  of  effect- 

7T 

ive  to  mean  value  of  wave,  that  is,  the  ratio  ,of  the  effective 
value  of  the  generated  e.m.f.  to  that  of  a  sine  wave  generated 
by  the  same  magnetic  flux  at  the  same  frequency. 

126 


SYNCHRONOUS  MACHINES  127 

The  form  factor  7  depends  upon  the  wave  shape  of  the  gener- 
ated e.m.f.  The  wave  shape  of  e.m.f.  generated  in  a  single  con- 
ductor on  the  armature  surface  is  identical  with  that  of  the  dis- 
tribution of  magnetic  flux  at  the  armature  surface  and  will  be 
discussed  more  fully  in  the  chapter  on  commutating  machines. 
The  wave  of  total  e.m.f.  is  the  sum  of  the  waves  of  e.m.f.  in  the 
individual  conductors,  added  in  their  proper  phase  relation,  as 
corresponding  to  their  relative  positions  on  the  armature  surface. 

4.  In  a   Y  or  star-connected  three-phase  machine,  if  EQ  = 
e.m.f.  per  circuit,  or  Y  or  star  e.m.f.,  E  =  E0  \/3  is  the  e.m.f. 
between  terminals  or  A  (delta)  or  ring  e.m.f.,  since  two  e.m.fs. 
displaced  by  60  degrees  are  connected  in  series  between  terminals 
(V3  =  2  cos  30°). 

In  a  A-connected  three-phase  machine,  the  e.m.f.  per  circuit 
is  the  e.m.f.  between  the  terminals,  or  A  e.m.f. 

In  a  F-connected  three-phase  machine,  the  current  per  circuit 
is  the  current  issuing  from  each  terminal,  or  the  line  current,  or 
Y  current. 

In  a  A-connected  three-pHase  machine,  if  J0  =  current  per 
circuit,  or  A  current,  the  current  issuing  from  each  terminal,  or 
the  line  or  F  current,  is 

/  =  /o  V3. 

Thus  in  a  three-phase  system,  A  current  and  e.m.f.,  and  F 
current  and  e.m.f.  (or  ring  and  start  current  and  e.m.f.  respect- 
ively), are  to  be  distinguished.  They  stand  in  the  proportion 
1  -  V3. 

As  a  rule,  when  speaking  of  current  and  of  e.m.f.  in  a  three- 
phase  system,  under  current  the  F  current  or  current  per  line,  and 
under  e.m.f.  the  A  e.m.f.  or  e.m.f.  between  lines  is  understood. 

5.  While  the  voltage  wave  of  a  single  conductor  has  the  same 
shape  as  the  distribution  of  the  magnetic  flux  at  the  armature 
circumference  and  so  may  differ  considerably  from  a  sine,  that 
is,  contain  pronounced  higher  harmonics,  the  terminal  voltage 
is  the  resultant  of  the  waves  of  many  conductors,  and,  especially 
with  a  distributed  armature  winding,  shows  the  higher  harmonics 
in  a  much  reduced  degree;  that  is,  the  resultant  is  nearer  sine 
shape,  and  some  harmonics  may  be  entirely  eliminated  in  the 
terminal  voltage  wave,  though  they  may  appear  in  the  voltage 
wave  of  a  single  conductor.     Thus,  for  instance,  in  a  three-phase 
F-connected  machine,  the  voltage  per  circuit,  or  F  voltage,  may 
contain  a  third  harmonic  and  multiples  thereof,  while  in  the 


128     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

voltage  between  the  terminals  this  third  harmonic  is  eliminated. 
The  voltage  between  the  terminals  is  the  resultant  of  two  Y 
voltages,  displaced  from  each  other  by  60  degrees.  Sixty  de- 
grees for  the  fundamental,  however,  is  3  X  60°  =  180°,  or  oppo- 
sition for  the  third  harmonic;  that  is,  the  third  harmonics  in 
those  two  Y  voltages,  which  combine  to  the  delta  or  terminal 
voltage,  are  opposite,  and  so  neutralize  each  other. 

Even  in  a  single  turn,  harmonics  existing  in  the  magnetic  field 
and  thus  in  the  single  conductor  can  be  eliminated  by  fractional 
pitch.  Thus,  if  the  pitch  of  the  armature  turn  is  not  180  de- 
grees, but  less  by  ->  the  e.m.fs.  generated  in  the  two  conductors 

n 

of  a  single  turn  are  not  exactly  in  phase,  but  differ  by  -  of  a  half 

fl> 

wave  for  the  fundamental,  and  thus  a  whole  half  wave  for  the 
nth  harmonic,  so  that  their  nth  harmonics  are  in  opposition  and 
thus  cancel.  Fractional  pitch  winding  of  a  "pitch  deficiency"  of 

-  thus  eliminates  the  nth  harmonic;  for  instance,  with  80  per 

^  •  . 

cent,  pitch,  the  fifth  harmonic  cannot  exist. 

In  this  manner  higher  harmonics  of  the  e.m.f.  wave  can  be 
reduced  or  entirely  eliminated,  though  in  general,  with  a  dis- 
tributed winding,  the  wave  shape  is  sufficiently  close  to  sine 
shape  without  special  precaution  being  taken  in  the  design. 

II.  Electromotive  Forces 

6.  In  a  synchronous  machine  we  have  to  distinguish  between 
terminal  voltage  E,  real  generated  e.m.f.  #1,  virtual  generated 
e.m.f.  EZ,  and  nominal  generated  e.m.f.  EQ. 

The  real  generated  e.m.f.  EI  is  the  e.m.f.  generated  in  the  alter- 
nator armature  turns  by  the  resultant  magnetic  flux,  or  mag- 
netic flux  interlinked  with  them,  that  is,  by  the  magnetic  flux 
passing  through  the  armature  core.  It  is  equal  to  the  terminal 
voltage  plus  the  e.m.f.  consumed  by  the  resistance  of  the  arma- 
ture, these  two  e.m.fs.  being  taken  in  their  proper  phase  relation; 
thus 

Ei  =  E  +  Ir, 

where  /  =  current  in  armature,  r  =  effective  resistance. 

The  virtual  generated  e.m.f.  E2  is  the  e.m.f.  which  would  be 
generated  by  the  flux  produced  by  the  field  poles,  or  flux  corre- 
sponding to  the  resultant  m.m.f.,  that  is,  the  resultant  of  the 


SYNCHRONOUS  MACHINES  129 

m.m.fs.  of  field  excitation  and  of  armature  reaction.  Since  the 
magnetic  flux  produced  by  the  armature,  or  flux  of  armature 
self-inductance,  combines  with  the  field  flux  to  the  resultant  flux, 
the  flux  produced  by  the  field  poles  does  not  pass  through  the 
armature  completely,  and  the  virtual  e.m.f.  and  the  real  gener- 
ated e.m.f.  differ  from  each  other  by  the  e.m.f.  of  armature  self- 
inductance;  but  the  virtual  generated  e.m.f.,  as  well  as  the  e.m.f. 
generated  in  the  armature  by  self-inductance,  have  no  real  and 
independent  existence,  but  are  merely  fictitious  components  of 
the  real  or  resultant  generated  e.m.f.  EI. 
The  virtual  generated  e.m.f.  is 

Ei  =  Et  +  jlx, 

where  x  is  the  self -inductive  armature  reactance,  and  the  e.m.f 
consumed  by  self-inductance  Ix  is  to  be  combined  with  the  real 
generated  e.m.f.  EI  in  the  proper  phase  relation. 

7.  The  nominal  generated  e.m.f.  EQ  is  the  e.m.f.  which  would 
be  generated  by  the  field  excitation  if  there  were  neither  self- 
inductance  nor  armature  reaction,  and  the  saturation  were  the 
same  as  corresponds  to  the  real  generated  e.m.f.  It  thus  does 
not  correspond  to  any  magnetic  flux,  and  has  no  existence  at  all, 
but  is  merely  a  fictitious  quantity,  which,  however,  is  very  useful 
for  the  investigation  of  alternators  by  allowing  the  combination 
of  armature  reaction  and  self-inductance  into  a  single  effect  by 
a  (fictitious)  self-inductance  or  synchronous  reactance  XQ.  The 
nominal  generated  e.m.f.  would  be  the  terminal  voltage  with 
open  circuit  and  load  excitation  if  the  saturation  curve  were  a 
straight  line. 

The  synchronous  reactance  XQ  is  thus  a  quantity  combining 
armature  reaction  and  self-inductance  of  the  alternator.  It  is 
the  only  quantity  which  can  easily  be  determined  by  experiment 
by  running  the  alternator  on  short  circuit  with  excited  field.  If 
in  this  case  IQ  =  current,  PQ  =  loss  of  power  in  the  armature 

coils,  EQ  =  e.m.f.  corresponding  to  the  field  excitation  at  open 
w  p 

circuit,  7—  =  ZQ  is  the  synchronous  impedance,  y^  =  r0  is  the 
-to  J-o 

effective  resistance  (ohmic  resistance  plus  load  losses),  and 
XQ  =  A/202  —  ro2  the  synchronous  reactance. 

In  this  feature  lies  the  importance  of  the  term  "  nominal 
generated  e.m.f."  EQ, 

E0  =  Ei  +  J!XQ,  =  E  +  (r  +  jx)  I 


130     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

the  terms  being  combined  in  their  proper  phase  relation.  In  a 
polyphase  machine,  these  considerations  apply  to  each  of  the 
machine  circuits  individually. 

III.  Armature  Reaction 

8.  The  magnetic  flux  in  the  field  of  an  alternator  under  load 
is  produced  by  the  resultant  m.m.f.  of  the  field  exciting  current 
and  of  the  armature  current.  It  depends  upon  the  phase  rela- 
tion of  the  armature  current.  The  e.m.f.  generated  by  the  field 
exciting  current  or  the  nominal  generated  e.m.f.  reaches  a  maxi- 
mum when  the  armature  coil  faces  the  position  midway  between 


FIG.  48. — Model  for  study  of  armature  reaction.     Armature  coils  in  position 
of  maximum  current. 

the  field  poles,  as  shown  in  Fig.  48,  A  and  A'.  Thus,  if  the 
armature  current  is  in  phase  with  the  nominal  generated  e.m.f., 
it  reaches  its  maximum  in  the  same  position  A,  A'  of  armature 
coil  as  the  nominal  generated  e.m.f.,  and  thus  magnetizes  the 
preceding,  demagnetizes  the  following  magnet  pole  (in  the  di- 
rection of  rotation)  in  an.  alternating-current  generator  A] 
magnetizes  the  following  and  demagnetizes  the  preceding  mag- 
net pole  in  a  synchronous  motor  A'  (since  in  a  generator  the 
rotation  is  against,  in  a  synchronous  motor  with  the  magnetic 
attractions  and  repulsions  between  field  and  armature).  In 
this  case  the  armature  current  neither  magnetizes  nor  demag- 
netizes the  field  as  a  whole,  but  magnetizes  the  one  side,  demag- 


SYNCHRONOUS  MACHINES 


131 


netizes  the  other  side  of  each  field  pole,  and  thus  merely  distorts 
the  magnetic  field. 

9.  If  the  armature  current  lags  behind  the  nominal  generated 
e.m.f.,  it  reaches  its  maximum  in  a  position  where  the  armature 
coil  already  faces  the  next  magnetic  pole,  as  shown  in  Fig.  48, 
B  and  Br,  and  thus  demagnetizes  the  field  in  a  generator  B, 
magnetizes  it  in  a  synchronous  motor  Bf. 

If  the  armature  current  leads  the  nominal  generated  e.m.f., 
it  reaches  its  maximum  in  an  earlier  position,  while  the  arma- 
ture coil  still  partly  faces  the  pre- 
ceding magnet  pole,  as  shown  in 
Fig.  48,  C  and  C",  and  thus  mag- 
netizes the  field  in  a  generator,  Fig. 
48,  C,  and  demagnetizes  it  in  a  syn- 
chronous motor  C'. 

With  non-inductive  load,  or  with 
the  current  in  phase  with  the  ter- 
minal voltage  of  an  alternating- 
current  generator,  the  current  lags 
behind  the  nominal  generated  e.m.f., 
due  to  armature  reaction  and  self- 
inductance,  and  thus  partly  de- 
magnetizes; that  is,  the  voltage  is 
lower  under  load  than  at  no  load 
with  the  same  field  excitation.  In 
other  words,  lagging  current  demag- 
netizes and  leading  current  magne- 
tizes the  field  of  an  alternating-cur- 
rent generator,  while  the  opposite  is 
the  case  with  a  synchronous  motor. 

10.  In  Fig.  49  let  OF  =  F  =  resultant  m.m.f.  of  field  exci- 
tation and  armature  current   (the  m.m.f.  of  the  field  excita- 
tion being  alternating  with  regard  to  the  armature  coil,  due  to 
its  rotation)  and  02  the  lag  of  the  current  /  behind  the  virtual 
e.m.f.  E<2,  generated  by  the  resultant  m.m.f. 

The  virtual  e.m.f.  E2  lags  in  time  90  degrees  behind  the  result- 
ant flux  of  OF,  and  is  thus  represented  by  OE2  in  Fig.  47,  and 
the  m.m.f.  of  the  armature  current  Fa  by  OFa,  lagging  by  angle 
02  behind  OE2.  The  resultant  m.m.f.  OF  is  the  diagonal  of  the 
parallelogram  with  the  component  m.m.fs.  OFa  =  armature 
m.m.f.  and  OFQ  =  total  impressed  m.m.f.  or  field  excitation,  as 


FIG.  49. — Diagram  of  m.m.fs.  in 
loaded  generator. 


132     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


sides,  and  from  this  construction  OFQ  is  found.  OF0  is  thus 
the  position  of  the  field  pole  with  regard  to  the  armature.  It 
is  trigonometrically, 


If  I  =  current  per  armature  turn  in  amperes  effective,  n  = 
number  of  turns  per  pole  in  a  single-phase  alternator,  the  arma- 
ture reaction  is  Fa  =  nl  ampere-turns  effective,  and  is  pulsating 
between  zero  and  nl  \/2. 

In  a  quarter-phase  alternator  with  n  turns  per  pole  and 
phase  in  series  and  I  amperes  effective  per  turn,  the  armature 
reaction  per  phase  is  nl  ampere-turns  effective  and  nl  \/2 
ampere-turns  maximum.  The  two  phases  magnetize  in  quad- 
rature, in  phase  and  in  space.  Thus,  at  the  time  t,  correspond- 
ing to  angle  6  after  the  maximum  of  the  first  phase,  the  m.m.f. 
in  the  direction  by  angle  6  behind  the  direction  of  the  magnetiza- 
tion of  the  first  phase  is  nl  \/2  cos2  6.  The  m.m.f.  of  the  second 
phase  is  nl  \/2  sin2  0;  thus  the  total  m.m.f.  or  the  armature 
reaction  Fa  =  nl  \/2,  and  is  constant  in  intensity,  but  revolves 
synchronously  with  regard  to  the  armature;  that  is,  it  is  station- 
ary with  regard  to  the  field. 

In  a  three-phaser  of  n  turns  in  series  per  pole  and  phase  and 
/  amperes  effective  per  turn,  the  m.m.f.  of  each  phase  is  nl  \/2 
ampere-turns  maximum;  thus  at  angle  6  in  positon  and  angle  0 
in  time  behind  the  maximum  of  one  phase; 

The  m.m.f.  of  this  phase  is 

nl  \/2  cos2  0. 

The  m.m.f.  of  the  second  phase  is 

nl  V2  cos2  (0  +  120)  =  n/V2  (  -  0.5  cos  0  -  0.5  \/3  sin  0)*. 
The  m.m.f.  of  the  third  phase  is 

nl  V2  cos2  (0  +  240)  =  nl  \/2  (  -  0.5  cos  0  +  0.5  \/3  sin  0)2. 
Thus  the  total  m.m.f.  or  armature  reaction, 

Fa  =  nl  \/2  (cos2  0  +  0.25  cos2  0  +  0.75  sin2  0  +  0.25  cos2  0 
+  0.75  sin2  0)  =  l.Snl  \/2, 

constant  in  intensity,  but  revolving  synchronously  with  regard 
to  the  armature,  that  is,  stationary  with  regard  to  the  field. 
These  values  of  armature  reaction  correspond  strictly  only  to 
the  case  where  all  conductors  of  the  same  phase  are  massed 


SYNCHRONOUS  MACHINES  133 

together  in  one  slot.  If  the  conductors  of  each  phase  are  dis- 
tributed over  a  greater  part  of  the  armature  surface,  the  values 
of  armature  reaction  have  to  be  multiplied  by  the  average  cosine 
of  the  total  angle  of  spread  of  each  phase. 

11.  The  single-phase  machine  thus  differs  from  the  poly- 
phase machines:  in  the  latter,  on  balanced  load,  the  armature 
reaction  is  constant,  while  in  the  single-phase  machine  the 
armature  reaction  and  thereby  the  resultant  m.m.f.  of  field  and 
armature  is  pulsating.  The  pulsation  of  the  resultant  m.m.f. 
of  the  single-phase  machine  causes  a  pulsation  of  its  magnetic 
field  under  load,  of  double  frequency,  which  generates  a  third 
harmonic  of  e.m.f.  in  the  armature  conductors.  In  machines 
of  high  armature  reaction,  as  steam-turbine-driven  single-phase 
alternators,  the  pulsation  of  the  magnetic  field  may  be  sufficient 
to  cause  serious  energy  losses  and  heating  by  eddy  currents, 
and  thus  has  to  be  checked.  This  is  usually  done  by  a  squirrel- 
cage  induction  machine  winding  in  the  field  pole  faces,  or  by 
short-circuited  conductors  laid  in  the  pole  faces  in  electrical 
space  quadrature  to  the  field  coils.  In 
these  conductors,  secondary  currents  Ei'_ 

of  double  frequency  are  produced 
which  equalize  the  resultant  m.m.f. 
of  the  machine. 


IV.  Self-inductance 

12.  The  effect  of  self -inductance  is  ^ 

similar  to  that  of  armature  reaction,    FlQ   50._Diagram  of  e  m  fs> 
and  depends  upon  the  phase  relation          in  loaded  generator, 
in  the  same  manner. 

If  EI  =  real  generated  voltage,  0i  =  lag  of  current  behind 
generated  voltage  EI,  the  magnetic  flux  produced  by  the  arma- 
ture current  I  is  in  phase  with  the  current,  and  thus  the  counter 
e.m.f.  of  self-inductance  is  in  quadrature  behind  the  current,  and 
therefore  the  e.m.f.  consumed  by  self-inductance  is  in  quadrature 
ahead  of  the  current.  Thus  in  Fig.  50,  denoting  OEi  =  EI  the 
generated  e.m.f.,  the  current  is  01  =  7;  lagging  61  behind  OEi, 
the  e.m.f.  consumed  by  self -inductance  OE  "i,  is  90  degrees  ahead 
of  the  current,  and  the  virtual  generated  e.m.f.  E2,  is  the  resultant 
of  OEi  and  OE'\.  As  seen,  the  diagram  of  e.m.f  s.  of  self -induc- 
tance is  similar  to  the  diagram  of  m.m.fs.  of  armature  reaction. 


134     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

13.  From  this  diagram  we  get  the  effect  of  load  and  phase  re- 
lation npon  the  e.m.f.  of  an  alternating-current  generator. 

Let  E  —  terminal  voltage  per  machine  circuit, 

7  =  current  per  machine  circuit, 
and      0  =  lag  of  the  current  behind  the  terminal  voltage. 

Let   r  =  resistance, 

x  =  reactance  of  the  alternator  armature. 


FIG.  51. — Diagram  showing  combined  effect  of  armature  reaction  and  arma- 
ture self-inductance. 


Then,  in  the  vector  diagram,  Fig.  51, 

OE  =  E,  the  terminal  voltage,  assumed  as  zero  vector. 
01  =  I,  the  current,  lagging  by  the  angle  EOI  =  0. 

_The  e.m.f.  consumed  by  resistance  is  OE \  =  Ir  in  phase  with 
01. 

The  e-m-i^  consumed  by  reactance  is  OEfz  —  Ix,  90  degrees 
ahead  of  01. 

The  real  generated  e.m.f.  is  found  by  combining  OE  and  OE\  to 


SYNCHRONOUS  MACHINES 


135 


The  virtual  generated  e.m.f.  is  OEi  and  OE'Z  combined  to 

=  E2. 


The  m.m.f.   required  to  produce  -this  e.m.f.  Ez  is  OF  =  F, 


Fa  I         E, 

FIG.  52. — Diagram  of  generator  e.m.fs.  and  m.m.fs.  for  non-reactive  load. 

90  deg.  ahead  of  OE2.     It  is  the  resultant  of  the  armature  m.m.f. 

or  armature  reaction  and  of  the  impressed  m.m.f.  or  field  excita- 
tion. The  armature  m.m.f.  is  in  phase  with  the  cur- 
rent 7,  and  is  nl  in  a  single-phase  machine,  n  \/2  / 
in  a  quarter-phase  machine,  1.5  \/2  nl  in  a  three- 
phase  machine,  if  n  =  number  of  armature  turns  per 
pole  and  phase.  The  m.m.f.  of  armature  reaction  is 
represented  in  the  diagram  by  OFa  of  Fa  in  phase 
with  01,  and  the  impressed  m.m.f.  or  field  excitation 
OFo  =  FQ  is  the  side  of  a  parallelogram  with  OF  as  diag- 
onal and  OFa  as  other  side;  or,  the  m.m.f.  consumed 
by  armature  reaction  is  represented  by  OF'a  =  Fa  in 
opposition  to  01.  Combining  OF'a  and  OF  gives  OFQ 
=  FQ,  the  field  excitation. 


F, 


FIG.  53. — Diagram  of  generator, 
e.m.fs.  and  m.m.fs.  for  lagging  reac- 
tive load.  Power-factor  0 . 50. 


FIG.  54. — Diagram  of  generator 
e.m.fs.  and  m.m.fs.  for  leading  reac- 
tive load.  Power-factor  0.50. 


136     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

In  Figs.  52,  53,  54  are  drawn  the  diagrams  for  0  =  zero  or 
non-inductive  load,  0  =  60  degrees,  or  60  degrees  lag  (inductive 
load  of  power-factor  0.50),  and  0  =  —  60  deg.,  or  60  deg.  lead 
(anti-inductive  load  of  power-factor  0.50). 

Thus  it  is  seen  that  with  the  same  terminal  voltage  E  a  much 
higher  field  excitation,  FQ,  is  required  with  inductive  load  than 
with  non-inductive  load,  while  with  anti-inductive  load  a 
much  lower  field  excitation  is  required.  With  open  circuit 

the  field  excitation  required  to  produce  the  terminal  voltage 

W 
E  would  be  ~r  F  =  FQQ,  or  less  than  the  field  excitation  J^o  with 

JCJQ 

non-inductive  load. 

Inversely,  with  constant  field  excitation,  the  voltage  of  an  al- 
ternator drops  with  non-inductive  load,  drops  much  more  with 
inductive  load,  and  drops  less,  or  even  rises,  with  anti-inductive 
load. 

V.  Synchronous  Reactance 

14.  In  general,  both  effects,  armature  self-inductance  and 
armature  reaction,  can  be  combined  by  the  term  "  synchronous 
reactance." 


FIG.    55. — Diagram  showing  effect 
of  synchronous  reactance. 


FIG.  56. — Diagram  of  generator 
e.m.f  s.  showing  affect  of  synchronous 
reactance  with  non-reactive  load. 


In  a  polyphase  machine,  the  synchronous  reactance  is  different, 
and  lower,  with  one  phase  only  loaded,  as  "  single-phase  synchro- 
nous reactance,"  than  with  all  phases  uniformly  loaded,  as  "  poly- 
phase synchronous  reactance."  The  resultant  armature  reac- 
tion of  all  phases  of  the  polyphase  machine  is  higher  than  that 
with  the  same  current  in  one  phase  only,  and  so  also  the  self- 


SYNCHRONOUS  MACHINES 


137 


inductive  flux,  as  resultant  flux  of  several  phases,  and  thus  rep- 
resents a  higher  synchronous  reactance. 

Let  r  =  effective  resistance, 

XQ  =  synchronous  reactance  of  armature,  as  discussed  in 
Section  II. 

Let  E  =  terminal  voltage, 
/  =  current, 

0  =  angle  of  lag  of  the  current  behind '  the  terminal  vol- 
tage. 

It  is  in  vector  diagram,  Fig.  55. 

OE  =  E  =  terminal  voltage  assumed  as  zero  vector.     01  = 


FIG.  57. — Diagram  of  generator 
e.m.fs.  showing  effect  of  synchronous 
reactance  with  lagging  reactive  load. 
6  =  60  degrees. 


FIG.  58. — Diagram  of  generator 
e.m.fs.  Showing  effect  of  synchro- 
nous reactance  with  leading  reactive 
load  6  =  —  60  degrees. 


I  =  current  lagging  by  the  angle  EOI  =  0  behind  the  terminal 
voltage. 

OE\  =  Ir  is  the  e.m.f.  consumed  by  resistance,  in  phase 
with  01  j  and  OE'o  =  Ix0  the  e.m.f.  consumed  by  the  synchronous 
reactance,  90  degrees  ahead  of  the_current  OI. 

OE'i  and  OE'Q  combined  give  OE'  =  E'  the  e.m.f.  consumed 
by  the  synchronous  impedance. 

Combining  OE'i,  OE'o,  OE  gives  the  nominal  generated  e.m.f. 
OEo  =  EQ,  corresponding  to  the  field  excitation  FQ. 

In  Figs.  56,  57,  58,  are  shown  the  diagrams  for  6  =  0  or  non- 
inductive  load,  6  =  60  degrees  lag  or  inductive  load,  and  &  — 
—  60  degrees  or  anti-inductive  load. 

Resolving  all  e.m.fs.  into  components  in  phase  and  in  quad- 
rature with  the  current,  or  into  power  and  reactive  components, 
in  symbolic  expression  we  have: 


138     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

the  terminal  voltage  E  =  E  cos  6  +  jE  sin  6 ; 
the  e.m.f.  consumed  by  resistance,  E\  =  ir; 
the  e.m.f.  consumed  by  synchronous  reactance,  E'0  =  +  jixQ, 
and  the  nominal  generated  e.m.f., 

E0  =  E  +  E\  +  E'Q  =  (E  cos  0  +  ir)  +  j  (E  sin  19  +  ix0) ; 

or,     since 

.  „  »    ,     ,      ,  /      power  current  \ 

cos  6  =  p  =  power-factor  of  the  load  (  =  -.   —. — ) 

\       total  current  / 

and 

q  =  \/l  —  p2  =  sin  0  =  inductance  factor  of  the  load 

(wattless  current\ 
total  current'    /  ' 
it  is 

Eo  =  (Ep  +»  +  j  (Eq  +  ix0), 


or,  in  absolute  values, 


Eo  =  V(Ep  +  ir)2  +  (Eq  +  ^0)2; 
hence, 

E  =  VE02  -  i2  (x0p  -  rq)2  -  i  (rp  -f  x^q). 

The  power  delivered  by  the  alternator  into  the  external  cir- 
cuit is 

P  =  iEp; 

that  is,  the  current  times  the  power  component  of  the  terminal 
voltage. 

The  electric  power  produced  in  the  alternator  armature  is 

Po  =  i(Ep  +  ir)-, 

that  is,  the  current  times  the  power  component  of  the  nomi- 
nal generated  e.m.f.,  or,  what  is  the  same  thing,  the  current  times 
the  power  component  of  the  real  generated  e.m.f. 

VI.  Characteristic  Curves  of  Alternating-current  Generator 

15.  In  Fig.  59  are  shown,  at  constant  terminal  voltage  E,  the 
values  of  nominal  generated  e.m.f.  E0,  and  thus  of  field  excitation 
FQ,  with  the  current  7  as  abscissas  and  for  the  three  conditions, 

1.  Non-inductive  load,  p  =  1,  q  =  0. 

2.  Inductive  load  of  0  =  60  degrees  lag,  p  =  0.5,  q  =  0.866. 

3.  Anti-inductive  load  of  —  6  =  60  degrees  lead,  p  =  0.5, 

q  =  -0.866. 


SYNCHRONOUS  MACHINES 


139 


The  values  r  =  0.1,  XQ  =  5,  E  =  1000,  are  assumed.  These 
curves  are  called  the  compounding  curves  of  the  synchronous 
generator. 

In  Fig.  60  are  shown,  at  constant  nominal  generated  e.m.f.  EQ, 
that  is,  at  constant  field  excitation  F0,  the  values  of  terminal  vol- 


E  = 


000 
£=5, 


'•*& 


z 


EO.F, 


500 


20     10     00     80    100     120    UO     160    180    200AMP. 

FIG.  59. — Synchronous  generator  compounding  curves. 

tage  E  with  the  current  I  as  abscissas  and  for  the  same  resistance 
and  synchronous  reactance  r  =  0.1,  XQ  =  5,  for  the  three  different 
conditions, 

1.  Non-inductive  load,  p  =  1,  q  =  0,  EQ  =  1127. 

2.  Inductive  load  of  60  degrees  lag, 

p  =  Q.5,  q  =  0.866,  E0  =  1458. 


140     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

3.  Anti-inductive  load  of  60  degrees  lead, 

p  =  0.5,  q  =  -0.866,  E0  =  628. 

The  values  of  EQ  (and  thus  of  FQ)  are  assumed  so  as  to  give 
E  =  1000  at  I  =  100.  These  curves  are  called  the  regulation 
curves  of  the  alternator,  or  the  field  characteristics  of  the  syn- 
chronous generator. 

In  Fig.  61  are  shown  the  load  curves  of  the  machine,  with  the 


40   60   80   100   120   140   160   180   200   220   240   260   280  AMP. 

FIG.  60. — Synchronous  generator  regulation  curves. 

current  I  as  abscissas  and  the  watts  output  as  ordinates  corre- 
sponding to  the  same  three  conditions  as  Fig.  60.  From  the 
field  characteristics  of  the  alternator  are  derived  the  open-cir- 
cuit voltage  of  1127  at  full  non-inductive  load  excitation,  which 
is  1.127  times  full-load  voltage;  the  short-circuit  current  225  at 
full  non-inductive  load  excitation,  which  is  2.25  times  full-load 
current;  and  the  maximum  output,  124  kw.,  at  full  non-induct- 
ive load  excitation,  which  is  1.24  times  rated  output,  at  775 
volts  and  160  amp.  It  depends  upon  the  point  on  the  field 


SYNCHRONOUS  MACHINES 


141 


characteristic  at  which  the  alternator  works,  whether  it  tends 
to  regulate  for,  that  is,  maintains,  constant  voltage,  or  constant 
current,  or  constant  power,  approximately. 


z 


L 


7 


\ 


H 


20    40    60    80   100   120   140   160   180   200   220   240   260   280  AMP. 

FIG.  61. — Synchronous  generator  load  curves. 

VII.  Synchronous  Motor 

16.  As  seen  in  the  preceding,  in  an  alternating-current  gen- 
erator the  field  excitation  required  for  a  given  terminal  voltage 
and  current  depends  upon  the  phase  relation  of  the  external 
circuit  or  the  load.  Inversely,  in  a  synchronous  motor  the 
phase  relation  of  the  current  into  the  armature  at  a  given  ter- 
minal voltage  depends  upon  the  field  excitation  and  the  load. 

Thus,  if  E  =  terminal  voltage  or  impressed  e.m.f.,  I  =  current, 
6  =  lag  of  current  behind  impressed  e.m.f.  in  a  synchronous 
motor  of  resistance  r  and  synchronous  reactance  XQ,  the  polar 
diagram  is  as  follows,  Fig.  62. 

OE  =  E  is  the  terminal  voltage  assumed  as  zero  vector.  The 
current  01  =  I  lags  by  the  angle  EOI  =  6. 

The  e.m.f.  consumed  by  resistance  isj9#'i  =  Ir.  The  e.m.f. 
consumed  by  synchronous  reactance,  OE'o  =  IxQ.  Thus,  com- 


142     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


bining  OE'i  and  OE'o  gives  OE',  the  e.m.f.  consumed  by  the 
synchronous  impedance.  The  e.m.f.  consumed  by  the  synchro- 
nous impedance  OE'  and  the  e.m.f.  consumed  by  the  nominal 
generated  or  counter  e.m.f.  of  the  synchronous  motor  OEo, 
combined,  give  the  impressed  e.m.f.  OE.  Hence  OEo  is  one 
side  of  a  parallelogram,  with  OE'  as  the  other  side,  and  OE  as 
diagonal.  OEoo  .(not  shown),  equal  and  opposite  OE0,  would 
thus  be  the  nominal  counter-generated  e.m.f.  of  the  synchronous 
motor. 

In  Figs.  63  to  65  are  shown  the  polar  diagrams  of  the  syn- 
chronous motor  for  6  =  0  deg.,  6  =  60  deg.,  6  =  —  60  deg.  It 
is  seen  that  the  field  excitation  has  to  be  higher  with  lead- 


d    E' 


FIG.    62. — Vector    diagram    of 
synchronous  motor. 


FIG.    63. — Vector    diagram    of 
synchronous  motor.     0=0 


ing  and  lower  with  lagging  current  in  a  synchronous  motor, 
while  the  opposite  is  the  case  in  an  alternating-current  generator. 

In  symbolic  representation,  by  resolving  all  e.m.fs.  into  power 
components  in  phase  with  the  current  and  wattless  components 
in  quadrature  with  the  current  i,  we  have: 

the  terminal  voltage,  E  =  E  cos  6  +  jE  sin  6  =  Ep  +  jEq; 
the  e.m.f.  consumed  by  resistance,  E/i  =  ir, 
and  the  e.m.f.  consumed  by  synchronous  reactance,  E'Q  =  +  jix0. 

Thus  the  e.m.f.  consumed  by  the  nominal  counter-generated 
e.m.f.  is 

Eo  =  E  -  E'i  -  E'Q  =  (E  cos  0  -  ir)  +  j  (E  sin  6  -  ixQ) 
=  (Ep  -  ir)  +  j(Eq  -  ixQ); 


SYNCHRONOUS  MACHINES  143 

or,  in  absolute  values, 


V(Ecos  e  -  ir)2  +  (Esin  6  -  ix0)* 


=  V(Ep-  ij 
hence, 

E  =  i  (rp  +  xQq)  ±  \/EQ2  —  i2  (x0p  —  rq)z. 

The  power  consumed  by  the  synchronous  motor  is 

P  =  iEp; 

that  is,  the  current  times  the  power  component  of  the  impressed 
e.m.f. 

/" 

Eo 


FIG.  64. — Vector  diagram  of  syn-  FIG.  65. — Vector"  diagram  of  synchronous 
chronous  motor.  6  =  60  deg.  motor.  0  =  —  60  degrees. 

The  mechanical  power  delivered  by  the  synchronous  motor 
armature  is 

Po  =  i(Ep-ir); 

that  is,  the  current  times  the  power  component  of  the  nominal 
counter-generated  e.m.f.  Obviously  to  get  the  available  mechan- 
ical power,  the  power  consumed  by  mechanical  friction  and  by 
molecular  magnetic  friction  or  hysteresis,  and  the  power  of  field 
excitation,  have  to  be  subtracted  from  this  value  P0. 

VIII.  Characteristic  Curves  of  Synchronous  Motor 

17.  In  Fig.  66  are  shown,  at  constant  impressed  e.m.f.  E, 
the  nominal  counter-generated  e.m.f.  EQ  and  thus  the  field 
excitation  FQ  required, 

1.  At  no  phase  displacement,  6  =  0,  or  for  the  condition  of 
minimum  input; 


144     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


2.  For  0  =  +  60,  or  60  deg.  lag:  p  =  0.5,  q  =  +  0.866,  and 

3.  For  0  =  -  60,  or  60  deg.  lead:  p  =  0.5,  q  =  -  0.866, 
with  the  current  I  as  abscissas,  the  constants  being 

r  =  0.1,  z0  =  5,  and  E  =  1000. 

These  curves  are  called  the  compounding  curves  of  the  syn- 
chronous motors. 

In  Fig.  67  are  shown,  with  the  power  output  PI  =  i  (Ep  —  ir) 
—  (iron  loss  and  friction)  as  abscissas,  and  the  same  constants 


1= 


E  = 
=0.1, 


000 
XQ= 


1100 


20    40    60    80    100   120   140   160   180   200 

FIG.  66. — Synchronous  motor  compounding  curves. 

r  =  0.1,  XQ  =  5,  E  =  1000,  and  constant  field  excitation  F0' 
that  is,  constant  nominal  counter-generated  e.m.f.  EQ  =  1109 
(corresponding  to  p  =  1,  #  =  0  at  7  =  100),  the  values  of 
current  I  and  power-factor  p.  As  iron  loss  is  assumed  3000 
watts,  as  friction  2000  watts.  Such  curves  are  called  load 
characteristics  of  the  synchronous  motor. 

18.  In  Fig.  68  are  shown,  with  constant  power  output  =  PO, 


SYNCHRONOUS  MACHINES 


145 


i  (Ep  —  ir),  and  the  same  constants,  r  =  0.1,  XQ  =  5,  E  =  1000, 
and  with  the  nominal  counter-generated  voltage  E0,  that  is,  field 
excitation  FQ,  as  abscissas,  the  values  of  current  /  for  the  four 
conditions, 

PO  =      5  kw.,  or  PI  =      0,  or  no  load, 

Po  =    50  kw.,  or  Pi  =    45  kw.,  or  half  load, 

Po  =    95  kw.,  or  Pi  =    90  kw.,  or  full  load, 

Po  =  140  kw.,  or  PI  =  135  kw.,  or  150  per  cent,  of  load. 


10    20   30    40    50    60    70    80    90   100   110   120   130   140 

FIG.  67.  —  Synchronous  motor  load  characteristics. 

Such  curves  are  called  phase  characteristics  of  the  synchronous 
motor. 
We  have 

Po  =  iEp  -  i*r. 
Hence, 

Po  +  iV 


P  = 

EQ  = 


-  p 


(Eq  -  ix0)2. 


Similar  phase  characteristics  exist  also  for  the  synchronous 
generator,  but  are  of  less  interest.  It  is  seen  that  on  each  of 
the  four-phase  characteristics  a  certain  field  excitation  gives 


146     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

minimum  current,  a  lesser  excitation  gives  lagging  current,  a 
greater  excitation  leading  current.  The  higher  the  synchronous 
reactance  XQ,  and  thus  the  armature  reaction  of  the  synchronous 
motor,  the  flatter  are  the  phase  characteristics;  that  is,  the  less 
sensitive  is  the  synchronous  motor  for  a  change  of  field  excitation 
or  of  impressed  e.m.f.  Thus  a  relatively  high  armature  reaction 
is  desirable  in  a  synchronous  motor  to  secure  stability,  that  is, 
independence  of  minor  fluctuations  of  impressed  voltage  or  of 
field  excitation. 

19.  The  theoretical  maximum  output  of  the  synchronous 
motor,  or  the  load  at  which  it  drops  out  of  step,  at  constant 
impressed  voltage  and  frequency  is,  even  with  very  high  armature 
reaction,  usually  far  beyond  the  heating  limits  of  the  machine. 


200     100     600     800     1000    1200    UOO    1600    1800    2000 

FIG.  66. — Synchronous  motor  phase  characteristics. 

The  actual  maximum  output  depends  on  the  drop  of  terminal 
voltage  due  to  the  increase  of  current,  and  on  the  steadiness  or 
uniformity  of  the  impressed  frequency,  thus  upon  the  individual 
conditions  of  operation,  but  is  as  a  rule  far  above  full  load. 

Hence,  by  varying  the  field  excitation  of  the  synchronous  motor 
the  current  can  be  made  leading  or  lagging  at  will,  and  the  syn- 
chronous motor  thus  offers  the  simplest  means  of  producing  out 
of  phase  or  wattless  currents  for  controlling  the  voltage  in  trans- 
mission lines,  compensating  for  wattless  currents  of  induction 
motors,  etc.  Synchronous  machines  used  merely  for  supplying 
wattless  currents,  that  is,  synchronous  motors  or  generators 
running  light,  with  over-excited  or  under-excited  field,  are  called 
synchronous  condensers.  They  are  used  as  exciters  for  induc- 
tion generators,  as  compensators  for  the  reactive  lagging  currents 


SYNCHRONOUS  MACHINES 


147 


of  induction  motors,  for  voltage  control  of  transmission  lines, 
etc.  Sometimes  they  are  called  "rotary  condensers"  or 
"dynamic  condensers"  when  used  only  for  producing  lead- 
ing currents. 

IX.  Magnetic  Characteristic  or  Saturation  Curve 

20.  The  dependence  of  the  generated  e.m.f.,  or  terminal 
voltage  at  open  circuit,  upon  the  field  excitation  is  called  the 
magnetic  characteristic,  or  saturation  curve,  of  the  synchronous 


1000      2000     3000     4000      5000 

FIG.  69. — Synchronous  generator  magnetic 


7000 

characteristics. 


machine.  It  has  the  same  general  shape  as  the  curve  of  mag- 
netic flux  density,  consisting  of  a  straight  part  below  saturation, 
a  bend  or  knee,  and  a  saturated  part  beyond  the  knee.  Gener- 
ally the  change  from  the  unsaturated  to  the  over-saturated  por- 
tion of  the  curve  is  more  gradual;  thus  the  knee  is  less  pronounced 
in  the  magnetic  characteristic  of  the  synchronous  machines,  since 
the  different  parts  of  the  magnetic  circuit  approach  saturation 
successively. 

The  dependence  of  the  terminal  voltage  upon  the  field  excita- 
tion, at  constant  full-load  current  through  the  amature  into  a 


148     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

non-inductive  circuit,  is  called  the  load  saturation  curve  of  the 

synchronous  machine.  It  is  a  curve  approximately  parallel  to 
the  no-load  saturation  curve,  but  starting  at  a  definite  value  of 
field  excitation  for  zero  terminal  voltage,  the  field  excitation 
required  to  maintain  full-load  current  through  the  armature 
against  its  synchronous  impedance. 

dF      dE 

The  ratio  -«•  -=-  ~FT 

r  Hi 

is  called  the  saturation  factor  s  of  the  machine.  It  gives  the  ratio 
of  the  proportional  change  of  field  excitation  required  for  a  change 

of  voltage.     The  quantity  5  =  1 is  called   the  percentage 

saturation  of  the  machine,  as  it  shows  the  approach  to  saturation. 

In  Fig.  69  is  shown  the  magnetic  characteristic  or  no-load 
saturation  curve  of  a  synchronous  generator,  the  load  satura- 
tion curve  and  the  no-load  saturation  factor,  assuming  E  = 
1000,  I  =  100  as  full-load  values. 

In  the  preceding  the  characteristic  curves  of  synchronous  ma- 
chines were  discussed  under  the  assumption  that  the  saturation 
curve  is  a  straight  line ;  that  is,  the  synchronous  machines  working 
below  saturation. 

21.  The  effect  of  saturation  on  the  characteristic  curves  of 
synchronous  machines  is  as  follows:  The  compounding  curve 
is  impaired  by  saturation;  that  is,  a  greater  change  of  field  excita- 
tion is  required  with  changes  of  load.  Under  load  the  magnetic 
density  in  the  armature  corresponds  to  the  true  generated  e.m.f. 
EI,  the  magnetic  density  in  the  field  to  the  virtual  generated 
e.m.f.  EI.  Both,  especially  the  latter,  are  higher  than  the  no- 
load  e.m.f.  or  terminal  voltage  E  of  the  generator,  and  thus  a 
greater  increase  of  field  excitation  is  required  in  the  presence  of 
saturation  than  in  the  absence  thereof.  In  addition  thereto,  due 
to  the  counter  m.m.f.  of  the  armature  current,  the  magnetic 
stray  field,  that  is,  that  magnetic  flux  which  leaks  from  field  pole 
to  field  pole  through  the  air,  increases  under  load,  especially  with 
inductive  load  where  the  armature  m.m.f.  directly  opposes  the 
field,  and  thus  a  still  further  increase  of  density  is  required  in  the 
field  magnetic  circuit  under  load.  In  consequence  thereof, 
at  high  saturation  the  load  saturation  curve  differs  more  from 
the  no-load  saturation  curve  than  corresponds  to  the  synchronous 
impedance  of  the  machine. 


SYNCHRONOUS  MACHINES 


149 


The  regulation  becomes  better  by  saturation;  that  is,  the 
increase  of  voltage  from  full  load  to  no  load  at  constant  field 
excitation  is  reduced,  the  voltage  being  limited  by  saturation. 
Owing  to  the  greater  difference  of  field  excitation  between  no  load 
and  full  load  in  the  case  of  magnetic  saturation,  the  improvement 
in  regulation  is  somewhat  reduced. 

X.  Efficiency  and  Losses 

22.  Besides  the  above  described  curves  the  efficiency  curves 
are  of  interest.  The  efficiency  of  alternators  and  synchronous 
motors  is  usually  so  high  that  a  direct  determination  by  measuring 
the  mechanical  power  and  the  electric  power  is  less  reliable  than 


10      20     30     4ff     50      60      TO     80     90     100  UO    120    130  140    150   160    170    ISO    150    200  KW. 

FIG.  70. — Synchronous  generator,  efficiency  and  losses. 

the  method  of  adding  the  losses,  and  the  latter  is  therefore  com- 
monly used. 

The  losses  consist  of  the  following:  the  resistance  loss  in  the 
armature;  the  resistance  loss  in  the  field  circuit;  the  hysteresis 
and  eddy  current  losses  in  the  magnetic  circuit;  the  friction  and 
windage  losses,  and  eventually  load  losses,  that  is,  losses  due  to 
eddy  currents  and  hysteresis  produced  by  the  load  current  in  the 
armature. 

The  resistance  loss  in  the  armature  is  proportional  to  the 
square  of  the  current,  I. 

The  resistance  loss  in  the  field  circuit  is  proportional  to  the 
square  of  the  field  excitation  current,  that  is,  the  square  of  the 

nominal  generated  or  counter-generated  e.m.f.,  EQ. 
10 


150     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

The  hysteresis  loss  is  proportional  to  the  1.6th  power  of  the 
real  generated  e.m.f.,  El  =  E  ±  Ir. 

The  eddy  current  loss  is  usually  proportional  to  the  square  of 
the  generated  e.m.f.,  E\. 

The  friction  and  windage  loss  is  assumed  as  constant. 

The  load  losses  vary  more  or  less  proportionally  to  the  square 
of  the  current  in  the  armature,  and  should  be  small  with  proper 
design.  They  can  often  be  represented  by  an  "effective"  arma- 
ture resistance. 

Assuming  in  the  preceding  example  a  friction  loss  of  2000  watts; 
an  iron  loss  of  3000  watts,  at  the  generated  e.m.f.  EI  =  1000;  a 


10  30  30  40   50  60  70 


1.00  UO  120  130  140  150  100  170  ISO  190  200  K.W, 


FIG.  71. — Synchronous  motor  efficiency  and  losses. 

resistance  loss  in  the  field  circuit  of  800  watts,  at  EQ  =  1000, 
and  a  load  loss  at  full  load  of  600  watts. 

The  loss  curves  and  efficiency  curves  are  plotted  in  Fig.  70 
for  the  generator,  with  the  current  output  at  non-inductive  load 
or  0  =  0  as  abscissas,  and  in  Fig.  71  for  the  synchronous  motor, 
with  the  mechanical  power  output  as  abscissas. 


XI.  Unbalancing  of  Polyphase  Synchronous  Machines 

23.  The  preceding  discussion  applies  to  polyphase  as  well  as 
single-phase  machines.  In  polyphase  machines  the  nominal 
generated  e.m.fs.  or  nominal  counter-generated  e.m.fs.  are  neces- 
sarily the  same  in  all  phases  (or  bear  a  constant  relation  to  each 
other).  Thus  in  a  polyphase  generator,  if  the  current  or  the 


SYNCHRONOUS  MACHINES  151 

phase  relation  of  the  current  is  different  in  the  different  branches, 
the  terminal  voltage  must  become  different  also,  more  or  less. 
This  is  called  the  unbalancing  of  the  polyphase  generator.  It  is 
due  to  different  load  or  load  of  different  inductance  factor  in  the 
different  branches. 

Inversely,  in  a  polyphase  synchronous  motor,  if  the  terminal 
voltages  of  the  different  branches  are  unequal,  due  to  an  unbal- 
ancing of  the  polyphase  circuit,  the  synchronous  motor  takes 
more  current  or  lagging  current  from  the  branch  of  higher  vol- 
tage, and  thereby  reduces  its  voltage,  and  takes  less  current  or 
leading  current1  from  the  branch  of  lower  voltage,  or  even 
returns  current  into  this  branch,  and  thus  raises  its  voltage. 
Hence  a  synchronous  motor  tends  to  restore  the  balance  of  an 
unbalanced  polyphase  system ;  that  is,  it  reduces  the  unbalancing 
of  a  polyphase  circuit  caused  by  an  unequal  distribution  or 
unequal  phase  relation  of  the  load  on  the  different  branches. 
To  a  less  degree  the  induction  motor  possesses  the  same  property. 

XII.  Starting  of  Synchronous  Motors 

24.  In  starting,  an  essential  difference  exists  between  the  single- 
phase  and  the  polyphase  synchronous  motor,  in  so  far  as  the  for- 
mer is  not  self-starting  but  has  to  be  brought  to  complete  syn- 
chronism, or  in  step  with  the  generator,  by  external  means  before 
it  can  develop  torque,  while  the  polyphase  synchronous  motor 
starts  from  rest  and  runs  up  to  synchronism  with  more  or  less 
torque. 

In  starting,  the  field  excitation  of  the  polyphase  synchronous 
motor  should  be  zero  or  very  low. 

The  starting  torque  is  due  to  the  magnetic  attraction  of  the 
armature  currents  upon  the  remanent  magnetism  left  in  the  field 
poles  by  the  currents  of  the  preceding  phase,  and  to  the  eddy 
currents  produced  therein. 

Let  Fig.  72  represent  the  magnetic  circuit  of  a  polyphase 
synchronous  motor.  The  m.m.f.  of  the  polyphase  armature 
currents  acting  upon  the  successive  projections  or  teeth  of  the 
armature,  1,  2,  3,  etc.,  reaches  a  maximum  in  them  successively; 
that  is,  the  armature  is  the  seat  of  a  m.m.f.  rotating  synchro- 
nously in  the  direction  of  the  arrow  A.  The  magnetism  in  the 

1  Since  with  lower  impressed  voltage  the  current  is  leading,  with  higher 
impressed  voltage  lagging,  in  a  synchronous  motor. 


152     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

face  of  the  field  pole  opposite  to  the  armature  projections  lags 
behind  the  m.m.f.,  due  to  hysteresis  and  eddy  currents,  and  thus 
is  still  remanent,  while  the  m.m.f.  of  the  projection  1  decreases, 
and  is  attracted  by  the  rising  m.m.f.  of  projection  2,  etc.,  or,  in 
other  words,  while  the  maximum  m.m.f.  in  the  armature  has  a 
position  a,  the  maximum  magnetism  in  the  field-pole  face  still 
has  the  position  b,  and  is  thus  attracted  toward  a,  causing  the 
field  to  revolve  in  the  direction  of  the  arrow  A  (or  with  a  station- 
ary field,  the  armature  to  revolve  in  the  opposite  direction  B). 

Lamination  of  the  field  poles  reduces  the  starting  torque 
caused  by  eddy  currents  in  the  field  poles,  but  increases  that 
caused  by  remanent  magnetism  or  hysteresis,  due  to  the  higher 
permeability  of  the  field  poles.  Thus  the  torque  per  volt-ampere 
input  is  approximately  the  same  in  either  case,  but  with  laminated 


i 

FIG.  72. — Magnetic  circuit  of  a  polyphase  synchronous  motor. 

poles  the  impressed  voltage  required  in  starting  is  higher  and  the 
current  lower  than  with  solid  field  poles.  In  either  case,  at  full 
impressed  e.m.f.  the  starting  current  of  a  synchronous  motor  is 
large,  since  in  the  absence  of  a  counter  e.m.f.  the  total  impressed 
e.m.f.  has  to  be  consumed  by  the  impedance  of  the  armature  cir- 
cuit. Since  the  starting  torque  of  the  synchronous  motor  is  due 
to  the  magnetic  flux  produced  by  the  alternating  armature  cur- 
rents, or  the  armature  reaction,  synchronous  motors  of  high 
armature  reaction  are  superior  in  starting  torque. 

Very  frequently  in  synchronous  motors  a  squirrel-cage  wind- 
ing is  used  in  the  field  pole  faces,  to  give  powerful  starting  torque 
by  the  induced  currents  therein,  on  the  induction  motor  principle. 
Such  squirrel-cage  winding  should  have  fairly  high  resistance  to 
start  well  from  rest,  but  low  resistance  to  give  powerful  syn- 
chronizing, that  is,  to  pull  its  load  promptly  into  synchronism. 


SYNCHRONOUS  MACHINES  153 

XIII.  Parallel  Operation 

25.  Any  alternator  can  be  operated  in  parallel,  or  synchronized 
with  any  other  alternator.  A  single-phase  machine  can  be  syn- 
chronized with  one  phase  of  a  polyphase  machine,  or  a  quarter- 
phase  machine  operated  in  parallel  with  a  three-phase  machine  by 
synchronizing  one  phase  of  the  former  with  one  phase  of  the 
latter.  Since  alternators  in  parallel  must  be  in  step  with  each 
other  and  have  the  same  terminal  voltage,  the  condition  of  satis- 
factory parallel  operation  is  that  the  frequency  of  the  machines 
is  identically  the  same,  and  the  field  excitation  such  as  would 
give  the  same  terminal  voltage.  If  this  is  not  the  case,  there  will 
be  cross  currents  between  the  alternators  in  a  local  circuit;  that 
is,  the  alternators  are  not  without  current  at  no  load,  and  their 
currents  under  load  are  not  of  the  same  phase  and  proportional 
to  their  respective  capacities.  The  cross  currents  between 
alternators  when  operated  in  parallel  can  be  wattless  currents  or 
power  currents. 

If  the  frequencies  of  two  alternators  are  identically  the  same, 
but  the  field  excitation  not  such  as  would  give  equal  terminal 
voltage  when  operated  in  parallel,  there  is  a  local  current  between 
the  two  machines  which  is  wattless  and  leading  or  magnetizing 
in  the  machine  of  lower  field  excitation,  lagging  or  demagnetiz- 
ing in  the  machine  of  higher  field  excitation.  At  load  this  watt- 
less current  is  superimposed  upon  the  currents  from  the  machines 
into  the  external  circuit.  In  consequence  thereof  the  current  in 
the  machine  of  higher  field  excitation  is  lagging  behind  the  cur- 
rent in  the  external  circuit,  the  current  in  the  machine  of  lower 
field  excitation  leads  the  current  in  the  external  circuit.  The 
currents  in  the  two  machines  are  thus  out  of  phase  with  each 
other,  and  their  sum  larger  than  the  joint  current,  or  current  in 
the  external  circuit.  Since  it  is  the  armature  reaction  of  leading 
or  lagging  current  which  makes  up  the  difference  between  the 
impressed  field  excitation  and  the  field  excitation  required  to  give 
equal  terminal  voltage,  it  follows  that  the  lower  the  armature 
reaction,  that  is,  the  closer  the  regulation  of  the  machines,  the 
more  sensitive  they  are  for  inequalities  or  variations  of  field 
excitation.  Thus,  too  low  armature  reaction  is  undesirable  for 
parallel  operation. 

With  identical  machines  the  changes  in  field  excitation  re- 
quired for  changes  of  load  must  be  the  same.  With  machines 


154     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

of  different  compounding  curves  the  changes  of  field  excitation 
for  varying  load  must  be  different,  and  such  as  correspond  to 
their  respective  compounding  curves,  if  wattless  currents  shall 
be  avoided.  With  machines  of  reasonable  armature  reaction 
the  wattless  cross  currents  are  small  even  with  relatively  great 
inequality  of  field  excitation.  Machines  of  high  armature  re- 
action have  been  operated  in  parallel  under  circumstances  where 
one  machine  was  entirely  without  field  excitation,  while  the  other 
carried  twice  its  normal  field  excitation,  with  wattless  currents, 
however,  of  the  same  magnitude  as  full-load  current. 

XIV.  Division  of  Load  in  Parallel  Operation 

26.  Much  more  important  than  equality  of  terminal  voltage 
before  synchronizing  is  equality  of  frequency.  Inequality  of 
frequency,  or  rather  a  tendency  to  inequality  of  frequency  (since 
by  necessity  the  machines  hold  each  other  in  step  or  at  equal 
frequency),  causes  cross  currents  which  transfer'energy  from  the 
machine  whose  driving  power  tends  to  accelerate  to  the  machine 
whose  driving  power  tends  to  slow  down,  and  thus  relieves  the 
latter  by  increasing  the  load  on  the  former.  Thus  these  cross 
currents  are  power  currents,  and  cause  at  no  load  or  light  load 
the  one  machine  to  drive  the  other  as  synchronous  motor,  while 
under  load  the  result  is  that  the  machines  do  not  share  the  load  in 
proportion  to  their  respective  capacities. 

The  speed  of  the  prime  mover,  as  steam  engine  or  turbine, 
changes  with  the  load.  The  frequency  of  alternators  driven 
thereby  must  be  the  same  when  in  parallel.  Thus  their  respect- 
ive loads  are  such  as  to  give  the  same  speed  of  the  prime  mover 
(or  rather  a  speed  corresponding  to  the  same  frequency) .  Hence 
the  division  of  load  between  alternators  connected  to  independent 
prime  movers  depends  almost  exclusively  upon  the  speed  regula- 
tion of  the  prime  movers.  To  make  alternators  divide  the  load 
in  proportion  to  their  capacities,  the  speed  regulation  of  their 
prime  movers  must  be  the  same;  that  is,  the  engines  or  turbines 
must  drop  in  speed  from  no  load  to  full  load  by  the  same  percent- 
age and  in  the  same  manner. 

If  the  regulation  of  the  prime  movers  is  not  the  same,  the  load 
is  not  divided  proportionally  between  the  alternators,  but  the 
alternator  connected  to  the  prime  mover  of  closer  speed  regula- 
tion takes  more  than  its  share  of  the  load  under  heavy  loads,  and 


SYNCHRONOUS  MACHINES  155 

less  under  light  loads.     Thus,  too  close  speed  regulation  of  prime 
movers  is  not  desirable  in  parallel  operation  of  alternators. 

XV.     Fluctuating  Cross  Currents  in  Parallel  Operation 

27.  In  alternators  operated  from  independent  prime  movers, 
it  is  not  sufficient  that  the  average  frequency  corresponding  to 
the  average  speed  of  the  prime  movers  be  the  same,  but  still 
more  important  that  the  frequency  be  the  same  at  any  instant, 
that  is,  that  the  frequency  (and  thus  the  speed  of  the  prime 
mover)  be  constant.  In  rotary  prime  movers,  as  turbines  or 
electric  motors,  this  is  usually  the  case;  but  with  reciprocating 
machines,  as  steam  engines,  the  torque  and  thus  the  speed  of 
rotation  rises  and  falls  periodically  during  each  revolution,  with 
the  frequency  of  the  engine  impulses.  The  alternator  con- 
nected with  the  engine  will  thus  not  have  uniform  frequency, 
but  a  frequency  which  pulsates,  that  is,  rises  and  falls.  The 
amplitude  of  this  pulsation  depends  upon  the  design  of  the  engine, 
the  momentum  of  its  fly-wheel,  and  the  action  of  the  engine 
governor. 

If  two  alternators  directly  connected  to  equal  steam  engines 
are  synchronized  so  that  the  moments  of  maximum  frequency 
coincide,  there  will  be  no  energy  cross  currents  between  the 
machines,  but  the  frequency  of  the  whole  system  rises  and  falls 
periodically.  In  this  case  the  engines  are  said  to  be  synchronized. 
The  parallel  operation  of  the  alternators  is  satisfactory  in  this 
case  provided  that  the  pulsations  of  engine  speeds  are  of  the 
same  size  and  duration;  but  apparatus  requiring  constant  fre- 
quency, as  synchronous  motors  and  rotary  converters,  when 
operated  from  such  a  system,  will  give  a  reduced  maximum  out- 
put, due  to  periodic  cross  currents  between  the  generators  of 
fluctuating  frequency  and  the  synchronous  motors  of  constant 
frequency,  and  in  an  extreme  case  the  voltage  of  the  whole  sys- 
tem will  be  caused  to  fluctuate  periodically.  Even  with  small 
fluctuations  of  engine  speed  the  unsteadiness  of  current  due  to  this 
source  is  noticeable  in  synchronous  motors  and  synchronous 
converters. 

If  the  alternators  happen  to  be  synchronized  in  such  a  position 
that  the  moment  of  maximum  speed  of  the  one  coincides  with 
the  moment  of  minimum  speed  of  the  other,  alternately  the 
one  and  then  the  other  alternator  will  run  ahead,  and  thus  there 


156     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

will  be  a  pulsating  power  cross  current  between  the  alternators, 
transferring  power  from  the  leading  to  the  lagging  machine, 
that  is,  alternately  from  the  one  to  the  other,  and  inversely, 
with  the  frequency  of  the  engine  impulses.  These  pulsating 
cross  currents  are  the  most  undesirable,  since  they  tend  to  make 
the  voltage  fluctuate  and  to  tear  the  alternators  out  of  synchro- 
nism with  each  other,  especially  when  the  conditions  are  favorable 
to  a  cumulative  increase  of  this  effect  by  what  may  be  called 
mechanical  resonance  (hunting)  of  the  engine  governors,  etc. 
They  depend  upon  the  synchronous  impedance  of  the  alternators 
and  upon  their  phase  difference,  that  is,  the  number  of  poles  and 
the  fluctuation  of  speed,  and  are  specially  objectionable  when 
operating  synchronous  apparatus  in  the  system. 

28.  Thus,  for  instance,  if  two  80-pole  alternators  are  directly 
connected  to  single-cylinder  engines  of  1  per  cent,  speed  varia- 
tion per  revolution,  twice  during  each  revolution  the  speed  will 
rise,  and  fall  twice;  and  consequently  the  speed  of  each  alternator 
will  be  above  average  speed  during  a  quarter  revolution.  Since 
the  maximum  speed  is  1/2  per  cent,  above  average,  the  mean 
speed  during  the  quarter  revolution  of  high  speed  is  1/4  per  cent, 
above  average  speed,  and  by  passing  over  20  poles  the  armature 
of  the  machine  will  during  this  time  run  ahead  of  its  mean  posi- 
tion by  1/4  per  cent,  of  20  or  1/20  pole,  that  is,  180/20  =  9  elec- 
trical space  degrees.  If  the  armature  of  the  other  alternator  at 
this  moment  is  behind  its  average  position  by  9  electrical  space 
degrees,  the  phase  displacement  between  the  alternator  e.m.fs.  is 
18  electrical  time  degrees;  that  is,  the  alternator  e.m.fs.  are 
represented  by  OEi  and  OEZ  in  Fig.  71,  and  when  running  in 
parallel  the  e.m.f.  OEf  =  E\E^  is  short  circuited  through  the 
synchronous  impedance  of  the  two  alternators.  . 

Since  E'  =  OE\  =  2  EI  sin  9  deg.  the  maximum  cross  current  is 

ffisin9deg.       0.156  ffi 
1    = = =  U.loo  1  o, 

20  20 

ET 

where  IQ  =  --  =  short-circuit  current  of  the  alternator  at  full- 

20 

load  excitation.  Thus,  if  the  short-circuit  current  of  the  alter- 
nator is  only  twice  full-load  current,  the  cross  current  is  31.2 
per  cent,  of  full-load  current.  If  the  short-circuit  current  is 
6  times  full-load  current,  the  cross  current  is  93.6  per  cent,  of 
full-load  current  or  practically  equal  to  full-load  current.  Thus 


SYNCHRONOUS  MACHINES  157 

the  smaller  the  armature  reaction,  or  the  better  the  regulation, 
the  larger  are  the  pulsating  cross  currents  between  the  alternators, 
due  to  the  inequality  of  the  rate  of  rotation  of  the  prime  movers. 
Hence  for  satisfactory  parallel  operation  of  alternators  connected 
to  steam  engines,  a  certain  amount  of  armature  reaction  is  de- 
sirable and  very  close  regulation  undesirable. 

By  the  transfer  of  energy  between  the  machines  the  pulsations 
of  frequency,  and  thus  the  cross  currents,  are  reduced  somewhat. 
Very  high  armature  reaction  is  objectionable  also,  since  it  reduces 
the  synchronizing  power,  that  is,  the  tendency  of  the  machines 
to  hold  each  other  in  step,  by  reducing  the  energy  transfer  be- 
tween the  machines.  As  seen  herefrom,  the  problem  of  parallel 
operation  of  alternators  is  al- 
most entirely  a  problem  of 
the  regulation  of  their  prime 

movers,       especially      steam  A ^^ 

engines. 

With  alternators  driven  by 
gas  engines,  the  problem  of 
parallel  operation  is  made 
more  difficult  by  the  more 
jerky  nature  of  the  gas-engine  ^  73._Phase  displacement  between 

impulse.      In    such   machines,  alternators  to  be  synchronized, 

therefore,  squirrel-cage  wind- 
ings in  the  field-pole  faces  are  commonly  used,  to  assist  synchron- 
izing by  the  currents  induced  in  this  short-circuited  winding,  on 
the  principle  of  the  induction  machine. 

From  Fig.  73  it  is  seen  that  the  e.m.f.  OEr  or  EiE2,  which 
causes  the  cross  current  between  two  alternators  in  parallel  con- 
nection, if  their  e.m.fs.  OEi  and  OE%  are  out  of  phase,  is  approxi- 
mately  in  quadrature  with  the  e.m.fs.  OE\  and  OE2  of  the 
machines,  if  these  latter  two  e.m.fs.  are  equal  to  each  other. 
The  cross  current  between  the  machines  lags  behind  the  e.m.f. 

producing  it,  OE* ',  by  the  angle  co,  where  tan  w  =  — ,  and  XQ  = 

7*0 

reactance,  r0  =  effective  resistance  of  alternator  armature.  The 
energy  component  of  this  cross  current,  or  component  in  phase 
with  OEfj  is  thus  in  quadrature  with  the  machine  voltages  OEi 
and  OE2,  that  is,  transfers  no  power  between  them.  The  power 
transfer  or  equalization  of  load  between  the  two  machines  takes 
place  by  the  wattless  or  reactive  component  of  cross  current, 


E' 


158     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

that  is,  the  component  which  is  in  quadrature  with  OE',  and  thus 
in  phase  with  one  and  in  opposition  with  the  other  of  the  machine 
e.m.fs.  OEi  and  OE^. 

29.  Hence,  machines  without  reactance  would  have  no  syn- 
chronizing power,  or  could  not  be  operated  in  parallel.  The 
theoretical  maximum  synchronizing  power  exists  if  the  reactance 
equals  the  resistance:  XQ  =  r0.  This  condition,  however,  cannot 
be  realized,  and  if  realized  would  give  a  dangerously  high  syn- 
chronizing power  and  cross  current.  In  practice,  XQ  is  always 
very  much  greater  than  ro,  and  the  cross  current  thus  practically 
in  quadrature  with  OE',  that  is,  in  phase  (or  opposition)  with 
the  machine  voltages,  and  is  consequently  an  energy-transfer 
current. 

If,  however,  alternators  are  operated  in  parallel  over  a  circuit 
of  appreciable  resistance,  as  two  stations  at  some  distance  from 
each  other  are  synchronized,  especially  if  the  resistance  between 
the  stations  is  non-inductive,  as  underground  cables,  with  alter- 
nators of  very  low  reactance,  as  turbo  alternators,  the  synchro- 
nizing power  may  be  insufficient.  In  this  case,  reactance  has  to  be 
inserted  between  the  stations,  to  lag  the  cross  current  and  thereby 
make  it  a  power-transferring  or  synchronizing  current. 

If,  however,  the  machine  voltages  OEi  and  OEZ  are  different 
in  value  but  approximately  in  phase  with  each  other,  the  voltage 
causing  cross  currents,  E\Ez ,  is  in  phase  with  the  machine  vol- 
tages and  the  crosscurrents  thus  in  quadrature  with  the  machine 
voltages  OEi  and  OE%,  and  hence  do  not  transfer  energy,  but  are 
wattless.  In  one  machine  the  cross  current  is  a  lagging  or 
demagnetizing,  and  in  the  other  a  leading  or  magnetizing, 
current. 

Hence  two  kinds  of  cross  currents  may  exist  in  parallel  opera- 
tion of  alternators — currents  transferring  power  between  the 
machines,  due  to  phase  displacement  between  their  e.m.fs.,  and 
wattless  currents  transferring  magnetization  between  the  ma- 
chines, due  to  a  difference  of  their  induced  e.m.fs. 

In  compound-wound  alternators,  that  is,  alternators  in  which 
the  field  excitation  is  increased  with  the  load  by  means  of  a 
series  field  excited  by  the  rectified  alternating  current,  it  is  al- 
most, but  not  quite,  as  necessary  as  in  direct-current  machines, 
when  operating  in  parallel,  to  connect  all  the  series  fields  in  paral- 
lel by  equalizers  of  negligible  resistance,  for  the  same  reason — to 
insure  proper  division  of  current  between  machines. 


SYNCHRONOUS  MACHINES  159 

XVI.  Higher  Frequency  Cross  Currents  between    Synchronous 

Machines 

30.  If  several  synchronous  machines  of  different  wave  shapes 
are  connected  into  the  same  circuit,  cross  currents  exist  between 
the  machines  of  frequencies  which  are  odd  multiples  of  the  circuit 
frequency,  that  is,  higher  harmonics  thereof.  The  machines  may 
be  two  or  more  generators,  in  the  same  or  in  different  stations, 
of  wave  shapes  containing  higher  harmonics  of  different  order, 
intensity  or  phase,  or  synchronous  motors  or  converters  of 
wave  shapes  different  from  that  of  the  system  to  which  they 
are  connected. 

The  intensity  of  these  cross  currents  is  the  difference  of  the 
corresponding  harmonics  of  the  machines  divided  by  the  impe- 
dance between  the  machines.  This  impedance  includes  the  self- 
inductive  reactance  of  the  machine  armatures.  The  reactance 
obviously  is  that  at  the  frequency  of  the  harmonic,  that  is,  if 
x  =  reactance  at  fundamental  frequency,  it  is  nx  for  the  nth 
harmonic. 

In  most  cases  these  cross  currents  are  very  small  and  negli- 
gible. With  machines  of  distributed  armature  winding,  the  in- 
tensity of  the  harmonic  is  low,  that  is,  the  voltage  nearly  a  sine 
wave,  and  with  machines  of  massed  armature  winding,  as  uni- 
tooth  alternators,  the  reactance  is  high.  These  cross  currents 
thus  usually  are  noticeable  only  at  no  load,  and  when  adjusting 
the  field  excitation  of  the  machines  for  minimum  current.  Thus 
in  a  synchronous  motor  or  converter,  at  no  load,  the  minimum 
current,  reached  by  adjusting  the  field,  while  small  compared  with 
full-load  current,  may  be  several  times  larger  than  the  minimum 
point  of  the  "  V"  curve  in  Fig.  68,  that  is,  the  value  of  the  energy 
current  supplying  the  losses  in  the  machine. 

It  is  only  in  the  parallel  operation  of  very  large  high-speed 
machines  (steam  turbine  driven  alternators)  of  high  armature 
reaction  and  very  low  armature  self-induction  that  such  high- 
frequency  cross  currents  may  require  consideration,  and  even  then 
only  in  three-phase  F-connected  generators  with  grounded 
neutral,  as  cross  currents  between  the  neutrals  of  the  machines. 
In  a  three-phase  machine,  the  voltage  between  the  terminals,  or 
delta  voltage,  contains  no  third  harmonic  or  its  multiple,  as  the 
third  harmonics  of  the  Y  voltage  neutralize  in  the  delta  voltage, 
and  such  a  machine,  with  a  terminal  voltage  of  almost  sine  shape, 


160     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

may  contain  a  considerable  third  harmonic  in  the  Y  voltage.  As 
the  three  Y  voltages  of  the  three-phase  system  are  120  degrees 
apart  in  phase,  their  third  harmonics  are  3  X  120  deg.  =  360  deg. 
apart,  or  in  phase  with  each  other,  from  the  main  terminals 
to  the  neutral,  and  by  connecting  the  neutrals  of  two  three- 
phase  machines  of  different  third  harmonics  with  each  other,  as 
by  grounding  the  neutrals,  a  cross  current  flows  between  the 
machines  over  the  neutral,  which  may  reach  very  high  values. 
Even  in  machines  of  the  same  wave  shape,  such  a  triple  frequency 
current  appears  between  the  machines  over  the  neutral,  when 
by  a  difference  in  field  excitation  a  difference  in  the  phase  of  the 
third  harmonic  is  produced.  It  therefore  is  often  undesirable  to 
ground  or  connect  together,  without  any  resistance,  the  neutrals 
of  three-phase  machines,  but  in  systems  of  grounded  neutral 
either  the  neutral  should  be  grounded  through  separate  resist- 
ances or  grounded  only  in  one  machine. 

XVII.  Short-circuit  Currents  of  Alternators 

31.  The  short-circuit  current  of  an  alternator  at  full-load 
excitation  usually  is  from  two  to  five  times  full-load  current, 
and  even  less  in  very  large  high-speed  steam  turbine  alternators. 
It  is 


where  EQ  =  nominal  generated  e.m.f.,  ZQ  =  synchronous  impe- 
dance of  alternator,  representing  the  combined  effect  of  arma- 
ture reaction  and  armature  self-inductance. 

In  the  first  moment  after  short  circuiting,  however,  the  current 
frequently  is  many  times  larger  than  the  permanent  short- 
circuit  current,  that  is, 


where  z  =  self-inductive  impedance  of  the  alternator. 

That  is,  in  the  first  moment  after  short  circuiting  the  poly- 
phase alternator  the  armature  current  is  limited  only  by  the  arma- 
ture self-inductance,  and  not  by  the  armature  reaction,  and 
some  appreciable  time  —  occasionally  several  seconds  —  elapses 
before  the  armature  reaction  becomes  effective. 

At  short  circuit,  the  magnetic  field  flux  is  greatly  reduced  by 
the  demagnetizing  action  of  the  armature  current,  and  the  gen- 


SYNCHRONOUS  MACHINES  161 

erated  e.m.f.  thereby  reduced  from  the  nominal  value  EQ  to  the 
virtual  value  Ez;  the  latter  is  consumed  by  the  armature  self- 
inductive  impedance  z,  or  self-inductive  reactance,  which  is 
practically  the  same  in  most  cases. 

The  armature  self-inductance  is  instantaneous,  since  the 
magnetic  field  rises  simultaneously  with  the  armature  current 
which  produces  it;  armature  reaction,  however,  requires  an  appre- 
ciable time  to  reduce  the  magnetic  flux  from  the  open-circuit 
value  to  the  much  lower  short-circuit  value,  since  the  magnetic 
field  flux  is  surrounded  by  the  field  exciting  coils,  which  act  as  a 
short-circuited  secondary  opposing  a  rapid  change  of  field  flux; 
that  is,  in  the  moment  when  the  short-circuit  current  starts  it 
begins  to  demagnetize  the  field,  and  the  magnetic  field  flux  there- 
fore begins  to  decrease;  in  decreasing,  however,  it  generates  an 
e.m.f.  in  the  field  coils,  which  opposes  the  change  of  field  flux, 
that  is,  increases  the  field  current  so  as  momentarily  to  main- 
tain the  full  field  flux  against  the  demagnetizing  action  of  the 
armature  reaction.  In  the  first  moment  the  armature  current 
thus  rises  to  the  value  given  by  the  e.m.f.  generated  by  the  full 
field  flux,  while  the  field  current  rises,  frequently  to  many  times 
its  normal  value  (hence,  if  circuit  breakers  are  in  the  field  circuit, 
they  may  open  the  circuit).  Gradually  the  field  flux  decreases, 
and  with  it  decrease  the  field  current  and  the  armature  current 
to  their  normal  values,  at  a  rate  depending  on  the  resistance 
and  the  inductance  of  the  field-exciting  circuit.  The  decrease 
in  value  of  the  field  flux  will  be  the  more  rapid  the  higher  the  re- 
sistance of  the  field  circuit,  the  slower  the  higher  the  inductance, 
that  is,  the  greater  the  magnetic  flux  of  the  machine.  Thus, 
the  momentary  short-circuit  current  of  the  machine  can  be  made 
to  decrease  somewhat  more  rapidly  by  increasing  the  resistance 
of  the  field  circuit,  that  is,  wasting  exciting  power  in  the  field 
rheostat. 

In  the  very  first  moment  the  short-circuit  current  waves  are 
unsymmetrical,  as  they  must  simultaneously  start  from  zero  in 
all  phases  and  gradually  approach  their  symmetrical  appear- 
ance, i.e.,  in  a  three-phase  machine  three  currents  displaced  by 
120  degrees.  Hereby  the  field  current  is  made  pulsating,  with  nor- 
mal or  synchronous  frequency,  that  is,  with  the  same  frequency 
as  the  armature  current.  This  full  frequency  pulsation  gradually 
dies  out  and  the  field  current  becomes  constant  with  a  polyphase 
short  circuit,  while  with  a  single-phase  short  circuit  it  remains 


162     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

pulsating  with  double  frequency,  due  to  the  pulsating  armature 
reaction.  In  a  polyphase  short  circuit  this  full  frequency  pul- 
sation due  to  the  unsymmetrical  starting  of  the  currents  is  inde- 
pendent of  the  point  of  the  wave  at  which  the  short  circuit 
starts,  since  the  resultant  asymmetry  of  all  the  polyphase  cur- 
rents is  the  same  regardless  of  the  point  of  the  wave  at  which  the 
circuit  is  closed.  In  a  single-phase  short  circuit,  however,  the 
full  frequency  pulsation  depends  on  the  point  of  the  wave  at  which 
the  circuit  is  closed,  and  is  absent  if  the  circuit  is  closed  at  that 
moment  at  which  the  short-circuit  current  would  pass  through 
zero. 

The  momentary  short-circuit  current  of  an  alternator  thus 
represents  one  of  the  few  cases  in  which  armature  self-induc- 
tance and  armature  reaction  do  not  act  in  the  same  manner,  and 
the  synchronous  reactance  can  be  split  into  two  components, 
thus,  XQ  =  x  -\-  x',  where  x  =  self-inductive  reactance,  which  is 
due  to  a  true  self-inductance,  and  x'  =  effective  reactance  of 
armature  reaction,  which  is  not  instantaneous. 

32.  In  machines  of  high  self-inductance  and  low  armature  re- 
action, as  high  frequency  alternators,  this  momentary  increase 
of  short-circuit  current  over  its  normal  value  is  negligible,  and 
moderate  in  machines  in  which  armature  reaction  and  self-in- 
ductance are  of  the  same  magnitude,  as  large  modern  multi- 
polar  low-speed  alternators.     In  large  high-speed  alternators  of 
high  armature  reaction  and  low  self -inductance,  as  steam  turbine 
alternators,  the  momentary  short-circuit  current  may  exceed  the 
permanent  value  ten  or  more  times.     With  such  large  currents 
magnetic  saturation  of  the  self-inductive  armature  circuit  still 
further  reduces  the  reactance  x,  that  is,  increases  the  current,  and 
in  such  cases  the  mechanical  shock  on  the  generator  becomes  so 
enormous  that  it  is  necessary  to  reduce  the  momentary  short-cir- 
cuit current  by  inserting  self-inductance,  that  is,  reactance  coils 
into  the  generator  leads,  or  by  specifically  designing  the  alterna- 
tor for  high  armature  reactance,  or  by  both. 

In  view  of  the  excessive  momentary  short-circuit  current,  it 
may  be*desirable  that  automatic  circuit  breakers  on  such  systems 
have  a  time  limit,  so  as  to  keep  the  circuit  closed  until  the  short- 
circuit  current  has  somewhat  decreased. 

33.  In  single-phase  machines,  and  in  polyphase  machines  in 
case  of  a  short  circuit  on  one  phase  only,  the  armature  reaction 
is  pulsating,  and  the  field  current  in  the  first  moment  after  the 


SYNCHRONOUS  MACHINES 


163 


short  circuit  therefore  pulsates,  with  double  frequency,  and 
remains  pulsating  even  after  the  permanent  condition  has  been 
reached.  The  double  frequency  pulsation  of  the  field  current 
in  case  of  a  single-phase  short  circuit  generates  in  the  armature 
a  third  harmonic  of  e.m.f.  The  short-circuit  current  wave  be- 
comes greatly  distorted  thereby,  showing  the  saw-tooth  shape 
characteristics  of  the  third  harmonic,  and  in  a  polyphase  machine 
on  single-phase  short  circuit,  in  the  phase  in  quadrature  with  the 
short-circuited  phase,  a  very  high  voltage  appears,  which  is  greatly 


Field  Current 


Armature  Current 


FIG.  74. — Three-phase  short-circuit  current  in  a  turbo-alternator. 

distorted  by  the  third  harmonic  and  may  reach  several  times 
the  value  of  the  open-circuit  voltage.  Thus,  with  a  single- 
phase  short  circuit  on  a  polyphase  system,  destructive  voltages 
may  appear  in  the  open-circuited  phase,  of  saw-tooth  wave  shape. 
Upon-  this  double  frequency  pulsation  of  the  field  current 
during  a  single-phase  short  circuit  the  transient  full  frequency 
pulsation  resulting  from  the  unsymmetrical  start  of  the  armature 
current  is  superimposed  and  thus  causes  a  difference  in  the  in- 
tensity of  successive  waves  of  the  double  frequency  pulsation, 


164     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

which  gradually  disappears  with  the  dying  out  of  the  transient 
full  frequency  pulsation,  and  depends  upon  the  point  of  the  wave 
at  which  the  short  circuit  is  closed,  and  thus  is  absent,  and  the 


Armature 
current. 


Field 
Current 


FIG.  75. — Single-phase    short-circuit    current    in     a    three-phase     turbo- 
alternator. 


.Armature 
current 


Field 
current 

52.5  amp. 


EJ  amp. 


FIG.  76. — Single-phase    short-circuit    current    in    a    three-phase    turbo- 
alternator. 

double  frequency  pulsation  symmetrical,  if  the  circuit  is  closed  at 
the  moment  when  the  short-circuit  current  should  be  zero. 

34.  As  illustration  is  shown,  in  Fig.  74,  the  oscillogram  of 


SYNCHRONOUS  MACHINES  165 

one  phase  of  the  three-phase  short  circuit  of  a  three-phase  turbo- 
alternator,  giving  the  unsymmetrical  start  of  the  armature 
currents  and  the  full  frequency  pulsation  of  the  field  current. 

In  Fig.  75  is  shown  a  single-phase  short  circuit  of  the  same 
machine,  in  which  the  circuit  is  closed  at  the  zero  value  of  the 
current;  the  current  wave  therefore  is  symmetrical,  and  the  field 
current  shows  only  the  double  frequency  pulsation  due  to  the 
single-phase  armature  reaction. 

In  Fig.  76  is  shown  another  single-phase  short  circuit,  in  which 
the  armature  current  wave  starts  unsymmetrical,  thus  giving  a 
transient  full  frequency  term  in  the  field  current.  Thus  in  the 
double  frequency  pulsation  of  the  field  current  at  first  large  and 
small  waves  alternate,  but  the  successive  waves  gradually  be- 
come equal  with  the  dying  out  of  the  full  frequency  term. 

In  Figs.  75  and  76  the  oscillogram  is  cut  off  by  the  open- 
ing of  the  circuit  breaker. 

For  further  discussion,  and  the  theoretical  investigation  of 
momentary  short-circuit  currents,  see  "Theory  and  Calculation 
of  Transient  Electric  Phenomena  and  Oscillations,"  Part  I, 
Chapters  XI  and  XII. 

For  further  discussion  of  the  terms  reactance,  armature  re- 
action and  field  excitation  and  their  relation,  see  "Theory  and 
Calculation  of  Electric  Circuits. " 


11 


B.  DIRECT-CURRENT  COMMUTATING  MACHINES 


I.  General 

35.  Commutating  machines  are  characterized  by  the  combina- 
tion of  a  continuously  excited  magnet  field  with  a  closed-circuit 
armature  connected  to  a  segmental  commutator.  According  to 
their  use,  they  can  be  divided  into  direct-current  generators 
which  transform  mechanical  power  into  electric  power,  direct- 
current  motors  which  transform  electric  power  into  mechanical 

power,  and  direct-current  con- 
verters which  transform  electric 
power  into  a  different  form  of 
electric  power.  Since  the  most 
important  class  of  the  latter 
are  the  synchronous  converters, 
which  combine  features  of  the 
synchronous  machines  with  those 
of  the  commutating  machines, 
they  shall  be  treated  in  a  sepa- 
rate chapter. 

By  the  excitation  of  their  mag- 
net fields,  commutating  machines 

are  divided  into  magneto  machines,  in  which  the  field  consists  of 
permanent  magnets;  separately  excited  machines;  shunt  machines, 
in  which  the  field  is  excited  by  an  electric  circuit  shunted  across 
the  machine  terminals,  and  thus  receives  a  small  branch  current 
at  full  machine  voltage,  as  shown  diagrammatically  in  Fig.  77; 
series  machines,  in  which  the  electric  field  circuit  is  connected  in 
series  with  the  armature,  and  thus  receives  the  full  machine  cur- 
rent at  low  voltage  (Fig.  78) ;  and  compound  machines,  excited  by 
a  combination  of  shunt  and  series  field  (Fig.  79).  .  In  compound 
machines  the  two  windings  can  magnetize  either  in  the  same  direc- 
tion (cumulative  compounding)  or  in  opposite  directions  (dif- 
ferential compounding).  Differential  compounding  has  been 
used  for  constant-speed  motors.  Magneto  machines  are  used 
only  for  very  small  sizes. 

166 


FIG.  77. — Shunt  machine. 


D.  C.  COMMUTATING  MACHINES 


167 


36.  By  the  number  of  poles  commutating  machines  are  divided 
into  bipolar  and  multipolar  machines.  Bipolar  machines  are 
mainly  used  in  small  sizes.  By  the  construction  of  the  armature, 
commutating  machines  are  divided  into  smooth-core  machines 
and  iron-clad  or  "toothed"  armature  machines.  In  the  smooth- 
core  machine  the  armature  winding  is  arranged  on  the  surface 
of  a  laminated  iron  core.  In  the  iron-clad  machine  the  arma- 
ture winding  is  sunk  into  slots.  The  iron-clad  type  has  the  ad- 
vantage of  greater  mechanical  strength,  but  the  disadvantage  of 
higher  self-inductance  in  commutation,  and  thus  requires  high- 
resistance,  carbon  or  graphite,  commutator  brushes.  The  iron- 
clad type  has  the  advantage  of  lesser  magnetic  stray  field,  due 


FIG.  78. — Series  machine. 


FIG.  79. — Compound  machine. 


to  the  shorter  gap  between  field  pole  and  armature  iron,  and  of 
lesser  magnet  distortion  under  load,  and  thus  can  with  carbon 
brushes  be  operated  with  constant  position  of  brushes  at  all  loads. 
In  consequence  thereof,  for  large  multipolar  machines  the  iron- 
clad type  of  armature  is  best  adapted;  the  smooth-core  type  is 
hardly  ever  used  nowadays. 

Either  of  these  types  can  be  drum  wound  or  ring  wound. 
The  drum  winding  has  the  advantage  of  lesser  self-inductance 
and  lesser  distortion  of  the  magnetic  field,  and  is  generally  less 
difficult  to  construct  and  thus  mostly  preferred.  By  the  arma- 
ture winding,  commutating  machines  are  divided  into  multiple- 
wound  and  series-wound  machines.  The  difference  between 
multiple  and  series  armature  winding,  and  their  modifications,  can 
best  be  shown  by  diagram. 


168     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

II.  Armature  Winding 

37.  Fig.  80  shows  a  six-pole  multiple  ring  winding,  and  Fig. 
81  a  six-polar  multiple  drum  winding.  As  seen,  the  armature 
coils  are  connected  progressively  all  around  the  armature  in 
closed  circuit,  and  the  connections  between  adjacent  armature 
coils  lead  to  the  commutator.  Such  an  armature  winding  has 
as  many  circuits  in  multiple,  and  requires  as  many  sets  of  com- 
mutator brushes,  as  poles.  Thirty-six  coils  are  shown  in  Figs. 
80  and  81,  connected  to  36  commutator  segments,  and  the 
two  sides  of  each  coil  distinguished  by  drawn  and  dotted  lines. 
In  a  drum-wound  machine,  usually  the  one  side  of  all  coils  forms 
the  upper  and  the  other  side  the  lower  layer  of  the  armature 
winding. 

Fig.  82  shows  a  six-pole  series  drum  winding  with  36  slots 
and  36  commutator  segments.  In  the  series  winding  the  circuit 
passes  from  one  armature  coil,  not  to  the  next  adjacent  armature 
coil  as  in  the  multiple  winding,  but  first  through  all  the  armature 
coils  having  the  same  relative  position  with  regard  to  the  magnet 
poles  of  the  same  polarity,  and  then  to  the  armature  coil  next  ad- 
jacent to  the  first  coil.  That  is,  all  armature  coils  having  the 
same  or  approximately  the  same  relative  position  to  poles  of  equal 
polarity  form  one  set  of  integral  coils.  Thus  the  series  winding 
has  only  two  circuits  in  multiple,  and  requires  two  sets  of  brushes 
only,  but  can  be  operated  also  with  as  many  sets  of  brushes  as 
poles,  or  any  intermediate  number  of  sets  of  brushes.  In  Fig.  82, 
a  series  winding  in  which  the  number  of  armature  coils  is  divisible 
by  the  number  of  poles,  the  commutator  segments  have  to  be 
cross  connected.  Therefore  this  form  of  series  winding  is  hardly 
ever  used.  The  usual  form  of  series  winding  is  the  winding  shown 
by  Fig.  83.  This  figure  shows  a  six-polar  armature  having  35 
coils  and  35  commutator  segments.  In  consequence  thereof  the 
armature  coils  under  corresponding  poles  which  are  connected 
in  series  are  slightly  displaced  from  each  other,  so  that  after  pass- 
ing around  all  corresponding  poles  the  winding  leads  symmetric- 
ally into  the  coil  adjacent  to  the  first  armature  coil.  Hereby  the 
necessity  of  commutator  cross  connections  is  avoided,  and  the 
winding  is  perfectly  symmetrical.  With  this  form  of  series 
winding,  which  is  mostly  used,  the  number  of  armature  coils  must 
be  chosen  to  follow  certain  rules.  Generally  the  number  of 
coils  is  one  less  or  one  more  than  a  multiple  of  the  number  of  poles. 


/).  C.  COMMUTATING  MACHINES 
s 


169 


FIG.  80. — Multiple  ring  armature  winding. 


FIG.  81. — Multiple  drum  full  pitch  winding. 


170     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


FIG.  82. — Series  drum  winding  with  cross-connected  commutator. 


FIG.  83. — Series  drum  winding. 


D.  C.  COMMUTATING  MACHINES 


171 


All  these  windings  are  closed-circuit  windings;  that  is,  starting 
at  any  point,  and  following  the  armature  conductor,  the  circuit 
returns  into  itself  after  passing  all  e.m.fs.  twice  in  opposite  direc- 
tion (thereby  avoiding  short  circuit).  An  instance  of  an  open- 
coil  winding  is  shown  in  Fig.  84,  a  series-connected  three-phase 
star  winding  similar  to  that  used  in  the  Thomson-Houston  arc 
machine.  Such  open-coil  windings,  however,  cannot  be  used  in 
commutating  machines.  They  are  generally  preferred  in  syn- 
chronous and  in  induction  machines. 


FIG.  84. — Open-circuit  three-phase  series  drum  winding. 

38.  By  leaving  space  between  adjacent  coils  of  these  windings 
a  second  winding  can  be  laid  in  between.  The  second  winding 
can  either  be  entirely  independent  from  the  first  winding,  that 
is,  each  of  the  two  windings  closed  upon  itself,  or  after  passing 
through  the  first  winding  the  circuit  enters  the  second  winding, 
and  after  passing  through  the  second  winding  it  reenters  the  first 
winding.  In  the  first  case  the  winding  is  called  a  double  spiral 
winding  (or  multiple  spiral  winding  if  more  than  two  windings 
are  used),  in  the  latter  case  a  double  reentrant  winding  (or 


172     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

multiple  reentrant  winding).  In  the  double  spiral  winding  the 
number  of  coils  must  be  even;  in  the  double  reentrant  winding, 
odd. 

Multiple  spiral  and  multiple  reentrant  windings  can  be  either 
multiple  or  series  wound;  that  is,  each  spiral  can  consist  either 
of  a  multiple  or  of  a  series  winding.  Fig.  85  shows  a  double 
spiral  multiple  ring  winding,  Fig.  86  a  double  spiral  multiple 
drum  winding,  Fig.  87  a  double  reentrant  multiple  drum  winding. 
As  seen  in  the  double  spiral  or  double  reentrant  multiple  wind- 
ing, twice  as  many  circuits  as  poles  are  in  multiple.  Thus  such 


FIG.  85. — Multiple  double  spiral  ring  winding. 

windings  are  mostly  used  for  large  low-voltage  machines,  but  as 
very  few  large  direct-current  generators  are  built  nowadays,  and 
alternating-current  generation  with  synchronous  converters  usu- 
ally preferred,  and  as  multiple  spiral  or  reentrant  windings  are 
inconvenient  in  synchronous  converters,  their  use  has  greatly 
decreased. 

39.  A  distinction  is  frequently  made  between  lap  winding  and 
wave  winding.  These  are,  however,  not  different  types;  but 
the  wave  winding  is  merely  a  constructive  modification  of  the 
series  drum  winding  with  single-turn  coil,  as  seen  by  comparing 


D.  C.  COMMUTATING  MACHINES 


173 


FIG.  86. — Multiple  double  spiral  full  pitch  winding. 


FIG.  87. — Multiple  double  re-entrant  drum  full  pitch  winding. 


174     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Figs.  88  and  89.  Fig.  88  shows  a  part  of  a  series  drum  winding 
developed.  Coils  C\  and  C2,  having  corresponding  positions  under 
poles  of  equal  polarity,  are  joined  in  series.  Thus  the  end  con- 
nection ah  of  coil  Ci  connects  by  cross  connection  be  and  cd  to  the 


FIG.  88. — Series  lap  winding. 

end  connection  de  of  coil  C%.  If  the  armature  coils  consist  of  a 
single  turn  only,  as  in  Fig.  86,  and  thus  are  open  at  6  and  d}  the 
end  connection  and  the  cross  connection  can  be  combined  by 
passing  from  a  in  coil  Ci  directly  to  c  and  from  c  directly  to  e  in 


FIG.  89. — Wave  winding. 


coil  C2;  that  is,  the  circuit  abcde  is  replaced  by  ace.     This  has 
the  effect  that  the  coils  are  apparently  open  at  one  side. 

Such  a  winding  has  been  called  a  wave  winding.  Only  series 
windings  with  a  single  turn  per  coil  can  be  arranged  as  wave 
windings,  while  windings  with  several  turns  per  coil  must  neces- 


D.  C.  COMMUTATING  MACHINES  175 

sarily  be  lap  or  coil  windings.  In  Fig.  90  is  shown  a  series  drum 
winding  with  35  coils  and  commutator  segments,  and  a  single 
turn  per  coil  arranged  as  wave  winding.  This  winding  may  be 
compared  with  the  35-coil  series  drum  winding  in  Fig.  83. 

40.  Drum  winding  can  be  divided  into  full-pitch  and  frac- 
tional-pitch windings.  In  the  full-pitch  winding  the  spread  of 
the  coil  covers  the  pitch  of  one  pole;  that  is,  each  coil  covers 


FIG.  90. — Series  drum  wave  winding. 

one-sixth  of  the  armature  circumference  in  a  six-pole  machine, 
etc.     In  a  fractional-pitch  winding  it  covers  less  or  more. 

Series  drum  windings  without  cross-connected  commutator  in 
which  thus  the  number  of  coils  is  not  divisible  by  the  number  of 
poles  are  necessarily  always  slightly  fractional  pitch;  but  gen- 
erally the  expression  " fractional-pitch  winding"  is  used  only  for 
windings  in  which  the  coil  covers  one  or  several  teeth  less  than 
correspond  to  the  pole  pitch.  Thus  the  multiple  drum  winding 
in  Fig.  81  would  be  a  fractional-pitch  winding  if  the  coils  spread 


176      ELEMENTS  OF  ELECTRICAL  ENGINEERING 

over  only  four  or  five  teeth  instead  of  over  six.     As  five-sixths 
fractional-pitch  multiple  drum  winding  it  is  shown  in  Fig.  91. 

Fractional-pitch  windings  have  the  advantage  of  shorter 
end  connections  and  less  self-inductance  in  commutation,  since 
commutation  of  corresponding  coils  under  different  poles  does 
not  take  place  in  the  same,  but  in  different,  slots,  and  the  flux 
of  self-inductance  in  commutation  is  thus  more  subdivided. 
Fig.  91  shows  the  multiple  drum  winding  of  Fig.  81  as  a  frac- 


FIG.  91. — Multiple  drum  five-sixth  fractional  pitch  winding. 

tional-pitch  winding  with  five  teeth  spread,  or  five-sixths  pitch. 
During  commutation  the  coils  a  b  c  d  e  f  commutate  simultane- 
ously. In  Fig.  81  these  coils  lie  by  twos  in  the  same  slots,  in 
Fig.  91  they  lie  in  separate  slots.  Thus,  in  the  former  case  the 
flux  of  self-inductance  interlinked  with  the  commutated  coil  is 
due  to  two  coils;  that  is,  twice  that  in  the  latter  case.  Frac- 
tional-pitch windings,  however,  have  the  disadvantage  of  reduc- 
ing the  width  of  the  neutral  zone,  or  zone  without  generated 
e.m.f.  between  the  poles,  in  which  commutation  takes  place, 


D.  C.  COMMUTATING  MACHINES  177 

since  the  one  side  of  the  coil  enters  or  leaves  the  field  before  the 
other.  Therefore,  in  commutating  machines  it  is  seldom  that 
a  pitch  is  used  that  falls  short  of  full  pitch  by  more  than  one 
or  two  teeth,  while  in  induction  and  synchronous  machines 
occasionally  as  low  a  pitch  as  50  per  cent,  is  used,  and  two-thirds 
pitch  is  frequently  employed. 

For  special  purposes,  as  in  single-phase  commutator  motors 
fractional-pitch  windings  are  sometimes  used. 

41.  Series  windings  find  their  foremost  application  in  machines 
with  small  currents,  or  small  machines  in  which  it  is  desirable 
to  have  as  few  circuits  as  possible  in  multiple,  and  in  machines 
in  which  it  is  desirable  to  use  only  two  sets  of  brushes,  as  in 
smaller  railway  motors.     In  multipolar  machines  with  many  sets 
of  brushes  a  series  winding  is  liable  to  give  selective  commutation; 
that  is,  the  current  does  not  divide  evenly  between  the  sets  of 
brushes  of  equal  polarity. 

Multiple  windings  are  used  for  machines  of  large  currents,  thus 
generally  for  large  machines,  and  in  large  low-voltage  machines 
the  still  greater  subdivision  of  circuits  afforded  by  the  multiple- 
spiral  and  the  multiple-reentrant  winding  is  resorted  to. 

To  resume,  then,  armature  windings  can  be  subdivided  into 

(a)  Ring  and  drum  windings. 

(6)  Closed-circuit  and  open-circuit  windings.  Only  the  former 
can  be  used  for  commutating  machines. 

(c)  Multiple  and  series  windings. 

(d)  Single-spiral,  multiple-spiral,  and  multiple-reentrant  wind- 

ings.    Either  of  these  can  be  multiple  or  series  windings. 

(e)  Full-pitch  and  fractional-pitch  windings. 

III.  Generated  E.M.FS. 

42.  The   formula   for  the  generation  of  e.m.f.  in  a  direct- 
current  machine,  as  discussed  in  the  preceding,  is 

e  = 

where  e  =  generated  e.m.f.,  /  =  frequency  =  number  of  pairs 
of  poles  X  hundreds  of  rev.  per  sec.,  n  =  number  of  turns  in 
series  between  brushes,  and  <£  =  magnetic  flux  passing  through 
the  armature  per  pole,  in  megalines. 

In  ring-wound  machines,  <f>  is  one-half  the  flux  per  field  pole, 
since  the  flux  divides  in  the  armature  into  two  circuits,  and  each 


178     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

armature  turn  incloses  only  half  the  flux  per  field  pole.  In  ring- 
wound  armatures,  however,  each  armature  turn  has  only  one  con- 
ductor lying  on  the  armature  surface,  or  face  conductor,  while  in  a 
drum-wound  machine  each  turn  has  two  face  conductors.  Thus, 
with  the  same .  number  of  face  conductors — that  is,  the  same 
armature  surface — the  same  frequency,  and  the  same  flux  per 
field  pole,  the  same  e.m.f.  is  generated  in  the  ring-wound  as  in 
the  drum-wound  armature. 

The  number  of  turns  in  series  between  brushes,  n,  is  one-half 
the  total  number  of  armature  turns  in  a  series-wound  armature, 

-  the  total  number  of  armature  turns  in  a  single-spiral  multiple- 
wound  armature  with  p  poles.  It  is  one-half  as  many  in  a  double- 
spiral  or  double-reentrant,  one-third  as  many  in  a  triple-spiral 
winding,  etc. 

By  this  formula,  from  frequency,  series  turns,  and  magnetic 
flux  the  e.m.f.  is  found,  or  inversely,  from  generated  e.m.f.,  fre- 
quency, and  series  turns  the  magnetic  flux  per  field  pole  is 
calculated: 

*--!-, 

4/n 

From  magnetic  flux,  and  section  and  lengths  of  the  different 
parts  of  the  rnagnetic  circuit,  the  densities  and  the  ampere- 
turns  required  to  produce  these  densities  are  derived,  and  as  the 
sum  of  the  ampere-turns  required  by  the  different  parts  of  the 
magnetic  circuit,  the  total  ampere-turns  excitation  per  field  pole 
is  found,  which  is  required  for  generating  the  desired  e.m.f. 

Since  the  formula  for  the  generation  of  direct-current  e.m.f 
is  independent  of  the  distribution  of  the  magnetic  flux,  or  its  wave 
shape,  the  total  magnetic  flux,  and  thus  the  ampere-turns  re- 
quired therefor,  are  independent  also  of  the  distribution  of 
magnetic  flux  at  the  armature  surface.  The  latter  is  of  impor- 
tance, however,  regarding  armature  reaction  and  commutation. 

IV.  Distribution  of  Magnetic  Flux 

43.  The  distribution  of  magnetic  flux  in  the  air  gap  or  at  the 
armature  surface  can  be  calculated  approximately  by  assuming 
the  density  at  any  point  of  the  armature  surface  as  proportional 
to  the  m.m.f.  acting  thereon,  and  inversely  proportional  to  the 
nearest  distance  from  a  field  pole.  Thus,  if  FQ  =  ampere-turns 


D.  C.  COMMUTATING  MACHINES 


179 


acting  upon  the  air  gap  between  armature  and  field  pole,  la  = 
length  of  air  gap,  from  iron  to  iron,  the  density  under  the  magnet 
pole,  that  is,  in  the  range  BC  of  Fig.  90,  is 


At  a  point  having  the  distance  lx  from  the  end  of  the  field  pole 
on  the  armature  surface,  the  distance  from  the  next  field  pole 
is  ld  =  Vk2  +  lx2,  and  the  density  thus,  approximately, 


B  C 

FIG.  92. — Distribution  of  mganetic  flux  under  a  single  pole. 


Herefrom  the  distribution  of  magnetic  flux  is  calculated  and 
plotted  in  Fig.  92,  for  a  single  pole  BC,  along  the  armature  sur- 
face A,  for  the  length  of  air  gap  la  =  1,  and  such  a  m.m.f.  as  to 


L 


FIG.  93. — Distribution  of  magnetic  force  and  flux  at  no  load. 

give  Bo  =  8000  under  the  field  pole;  that  is,  for  /0  =  6400  or 
HQ  =  8000. 

Around  the  surface  of  the  direct-current  machine  armature, 
alternate  poles  follow  each  other.  Thus  the  m.m.f.  is  constant 
only  under  each  field  pole,  but  decreases  in  the  space  between 
the  field  poles,  from  C  to  E  in  Fig.  93,  from  full  value  at  C  to 
full  value  in  opposite  direction  at  E.  The  point  D  midway 


180     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

between  C  and  E}  at  which  the  m.m.f.  of  the  field  equals  zero, 
is  called  the  "neutral."  The  distribution  of  m.m.f.  of  field 
excitation  is  thus  given  by  the  line  F  in  Fig.  91.  The  distribu- 
tion of  magnetic  flux  as  shown  in  Fig.  91  by  BQ  is  derived  by 
the  formula 

4irF 


B 


10  I 


where 


This  distribution  of  magnetic  flux  applies  only  to  the  no-load 
condition.  Under  load,  that  is,  if  the  armature  carries  current, 
the  distribution  of  flux  is  changed  by  the  m.m.f.  of  the  armature 
current,  or  armature  reaction. 


J 


FIG.  94. — Distribution  of  flux  with  current  in  the  armature. 

44.  Assuming  the  brushes  set  at  the  middle  points  between 
adjacent  poles,  D  and  G,  Fig.  94,  the  m.m.f.  of  the  armature  is 
maximum  at  the  point  connected  with  the  commutator  brushes, 
in  this  case  at  the  points  D  and  G}  and  gradually  decreases  from 
full  value  at  D  to  equal  but  opposite  value  at  G,  as  shown  by 
the  line  -Fa  in  Fig.  94,  while  the  line  FQ  gives  the  field  m.m.f. 
or  impressed  m.m.f. 

If  n  =  number  of  turns  in  series  between  brushes  per  pole, 
i  =  current  per  turn,  the  armature  reaction  is  Fa  =  ni  ampere- 
turns.  Adding  Fa  and  F0  gives  the  resultant  m.m.f.  F,  and  there- 
from the  magnetic  distribution: 


B  = 


The  latter  is  shown  as  line 


10  ld 
in  Fig.  94. 


D.  C.  COMMUTATING  MACHINES  181 

With  the  brushes  set  midway  between  adjacent  field  poles, 
the  armature  m.m.f.  is  additive  on  one  side  and  subtractive  on 
the  other  side  of  the  center  of  the  field  pole.  Thus  the  magnetic 
intensity  is  increased  on  one  side  and  decreased  on  the  other. 
The  total  m.m.f.,  however,  and  thus,  neglecting  saturation,  the 
total  flux  entering  the  armature,  are  not  changed.  Thus,  arma- 
ture reaction,  with  the  brushes  midway  between  adjacent  field 
poles,  acts  distorting  upon  the  field,  but  neither  magnetizes  nor 
demagnetizes,  if  the  field  is  below  saturation. 

The  distortion  of  the  magnetic  field  takes  place  by  the  arma- 
ture ampere-turns  beneath  the  pole,  or  from  B  to  C.  Thus,  if 
T  =  pole  arc,  that  is,  the  angle  covered  by  pole  face  (two  poles 
or  one  complete  period  being  denoted  by  360  degrees),  the  dis- 

rFa 
torting  ampere-turns  of  the  armature  reaction  are 


As  seen,  in  the  assumed  instance,  Fig.  94,  where  F 


the  m.m.f.  at  the  two  opposite  pole  corners,  and  thus  the  mag- 
netic densities,  stand  in  the  proportion  1  to  3.  As  seen,  the 
generated  e.m.f.  is  not  changed  by  the  armature  reaction,  with 
the  brushes  set  midway  between  the  field  poles,  except  by  the 
sftiall  amount  corresponding  to  the  flux  entering  beyond  D  and 
G,  that  is,  shifted  beyond  the  position  of  brushes.  At  D,  how- 
ever, the  flux  still  enters  the  armature,  depending  in  intensity 
upon  the  armature  reaction;  and  thus  with  considerable  arma- 
ture reaction  the  brushes  when  set  at  this  point  are  liable  to 
spark  by  short-circuiting  an  active  e.m.f.  Therefore,  under  load, 
the  brushes  are  shifted  toward  the  following  pole,  that  is,  toward 
the  direction  in  which  the  zero  point  of  magnetic  flux  has  been 
shifted  by  the  armature  reaction. 

45.  In  Fig.  95,  the  brushes  are  assumed  as  shifted  to  the  cor- 
ner of  the  next  pole,  E  respectively  B.  In  consequence  thereof, 
the  subtractive  range  of  the  armature  m.m.f.  is  larger  than  the 
additive,  and  the  resultant  m.m.f.  F  =  F0  +  Fa  is  decreased; 
that  is,  with  shifted  brushes  the  armature  reaction  demagnet- 
izes the  field.  The  demagnetizing  armature  ampere-turns  are 

f~ir> 

PM  =  7^TfFa.    That  is,  if  TI  =  angle  of  shift  of  brushes  or  angle 

of  lead  (  =  GB  in  Fig.  95),  assuming  the  pitch  of  two  poles  =  360 
degrees,  the  demagnetizing  component  of  armature  reaction  is 
2  T  V  V 

*";  the  distorting  component  is  ^-^.,  where  T  =  pole  arc. 

loU  loU 

12 


182     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Thus,  with  shifted  brushes  the  field  excitation  has  to  be  in- 
creased under  load  to  maintain  the  same  total  resultant  m.m.f., 

that  is,  the  same  total  flux  and  generated  e.m.f.     Hence,  in 

*?    jf        ff 

Fig.  95  the  field  excitation  F0  has  been  assumed  by     **a  =  - 

loU          o 

larger  than  in  the  previous  figures,  and  the  magnetic  distribution 
BI  plotted  for  these  values. 


J 


FIG.  95. — Distribution  of  flux  with  current  in  the  armature  and  brushes 
shifted  from  the  magnetic  neutral. 


V.  Effect  of  Saturation  on  Magnetic  Distribution 

46.  The  preceding  discussion  of  Figs.  92  to  95  omits  the  effect 
of  saturation.  That  is,  the  assumption  is  made  that  the  mag- 
netic materials  near  the  air  gap,  as  pole  face  and  armature  teeth, 
are  so  far  below  saturation  that  at  the  demagnetized  pole  corner 
the  magnetic  density  decreases,  at  the  strengthened  pole  corner 
increases,  proportionally  to  the  m.m.f. 

The  distribution  of  m.m.f.  obviously  is  not  affected  by  satu- 
ration, but  the  distribution  of  magnetic  flux  is  greatly  changed 
thereby.  To  investigate  the  effect  of  saturation,  in  Figs.  96  to 
99  the  assumption  has  been  made  that  the  air  gap  is  reduced  to 
one-half  its  previous  value,  la  =  0.5,  thus  consuming  only  one- 
half  as  many  ampere-turns,  and  the  other  half  of  the  ampere- 
turns  are  consumed  by  saturation  of  the  armature  teeth.  The 
length  of  armature  teeth  is  assumed  as  3.2,  and  the  space  filled 
by  the  teeth  is  assumed  as  consisting  of  one-third  of  iron  and 
two-thirds  of  non-magnetic  material  (armature  slots,  ventilating 
ducts,  insulation  between  laminations,  etc.). 


D.  C.  COMMUTATING  MACHINES  183 

In  Figs.  96,  97,  98,  99,  curves  are  plotted  corresponding  to 
those  in  Figs.  92,  93,  94,  and  95.  As  seen,  the  spread  of  mag- 
netic flux  at  the  pole  corners  is  greatly  increased,  but  farther 
away  from  the  field  poles  the  magnetic  distribution  is  not 
changed. 

47.  The  magnetizing,  or  rather  demagnetizing,  effect  of  the 
load  with  shifted  brushes  is  not  changed.  The  distorting  effect 


FIG.  96. — Flux  distribution  under  a  single  pole. 

of  the  load  is,  however,  very  greatly  decreased,  to  a  small  per- 
centage of  its  previous  value,  and  the  magnetic  field  under  the 
field  pole  is  very  nearly  uniform  under  load. 

The  reason  is:  Even  a  very  large  increase  of  m.m.f.  does  not 
much  increase  the  density,  the  ampere-turns  being  consumed  by 
saturation  of  the  iron,  and  even  with  a  large  decrease  of  m.m.f. 
the  density  is  not  decreased  much,  since  with  a  small  decrease 


'I/       •    \\      L 


FIG.  97. — Distribution  of  flux  and  m.m.f.   at  no  load. 

of  density  the  ampere-turns  consumed  by  the  saturation  of  the 
iron  become  available  for  the  air  gap. 

Thus,  while  in  Fig.  95  the  densities  at  the  center  and  the  two 
pole  corners  of  the  field  pole  are  8000,  12,000,  and  4000,  with  the 
saturated  structure  in  Fig.  99  they  are  8000,  9040,  and  6550. 

At  or  near  the  theoretical  neutral,  however,  the  saturation  has 
no  effect. 

That  is,  saturation  of  the  armature  teeth  affords  a  means  of 


184     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


reducing  the  distortion  of  the  magnetic  field,  or  the  shifting  of 
flux  at  the  pole  corners,  and  is  thus  advantageous  for  machines 
which  shall  operate  over  a  wide  range  of  load  with  fixed  position 
of  brushes,  if  the  brushes  are  shifted  near  to  the  next  following 
pole  corner. 

Fo 


J 


FIG.  98. — Distribution  of  flux  and  m.m.f.  at  load,  with  Brushes  at  neutral. 

It  offers  no  direct  advantage,  however,  for  machines  corn- 
mutating  with  the  brushes  midway  between  the  field  poles,  as 
converters. 

An  effect  similar  to  saturation  in  the  armature  teeth  is  produced 


J 


L 


FIG.  99. — Distribution  of  flux  and  m.m.f.  at  load,  with  brushes  shifted  to 

next  pole  corner. 

by  saturation  of  the  field  pole  face,  or  more  particularly,  satura- 
tion of  the  pole  corners  of  the  field. 

VI.  Effect  of  Commutating  Poles 

48.  With  the  commutator  brushes  of  a  generator  set  midway 
between  the  field  poles,  as  in  Fig.  94,  the  m.m.f.  of  armature  reac- 


D.  C.  COMMUTATING  MACHINES  185 

tion  produces  a  magnetic  field  at  the  brushes.  The  e.m.f.  gener- 
ated by  the  rotation  of  the  armature  through  this  field  opposes 
the  reversal  of  the  current  in  the  short-circuited  armature  coil 
under  the  brush,  and  thus  impairs  commutation.  If  therefore 
the  commutation  constants  of  the  machines  are  not  abnormally 
good — high  field  strength,  low  armature  reaction,  low  self-in- 
ductance and  frequency  of  commutation — the  machine  does  not 
commutate  satisfactorily  under  load,  with  the  brushes  midway 
between  the  field  poles,  and  the  brushes  have  to  be  shifted  to  the 
edge  of  the  next  field  poles,  as  shown  in  Fig.  95,  until  the  fringe 
of  the  magnetic  flux  of  the  field  poles  reverses  the  armature  reac- 
tion and  so  generates  an  e.m.f.  in  the  armature  coil,  which  re- 
verses the  current  and  thus  acts  as  commutating  flux.  The 
commutating  e.m.f.  and  therefore  the  commutating  flux  should 
be  proportional  to  the  current  which  is  to  be  reversed,  that  is, 
to  the  load.  The  magnetic  flux  of  the  field  pole  of  a  shunt  or 
compound  machine,  however,  decreases  with  increasing  load  at 
the  pole  corners  toward  which  the  brushes  are  shifted,  by  the 
demagnetizing  action  of  the  armature  reaction,  and  the  shift  of 
brushes  therefore  has  to  be  increased  with  the  load,  from  nothing 
at  no  load.  At  overload,  the  pole  corners  towarp!  which  the 
brushes  are  shifted  may  become  so  far  weakened  that  even 
under  the  pole  not  sufficient  reversing  e.m.f.  is  generated,  and 
satisfactory  commutation  ceases,  that  is,  the  sparking  limit  is 
reached. 

In  general,  however,  varying  the  brush  shift  with  the  load  is 
not  permissible,  and  with  rapidly  fluctuating  load  not  feasible, 
and  therefore  the  brushes  are  set  permanently  at  a  mean  shift. 
In  this  case,  however,  instead  of  increasing  proportionally  with 
the  load,  the  commutating  field  is  maximum  at  no  load,  and 
gradually  decreases  with  increase  of  load,  and  is  correct  only 
at  one  particular  load.  At  constant  shift  of  the  brushes,  the 
commutation  of  the  constant  potential  machine,  direct-current 
generator  or  motor,  is  best  at  a  certain  load,  and  usually  becomes 
poorer  at  lighter  or  heavier  loads,  and  ultimately  becomes  bad 
by  inductive  sparks  due  to  insufficient  commutating  flux.  In 
machines  in  which  very  good  commutating  constants  cannot  be 
secured,  as  in  large  high-speed  machines  (steam  turbine  driven 
direct-current  generators) ,  this  may  lead  to  bad  sparking  or  even 
flashing  over  at  sudden  overloads  as  well  as  when  throwing  off 
full  load. 


186     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

49.  This  has  led  to  the  development  of  the  commutating  pole, 
also  called  interpole,  that  is,  a  narrow  magnetic  pole  located 
between  the  main  poles  at  the  point  of  the  armature  surface, 
at  which  commutation  occurs,  and  excited  so  as  to  produce  a 
commutating  flux  proportional  to  the  load,  and  thus  giving  the 
required  commutating  field  at  all  loads.  Such  machines  then 
give  no  inductive  sparking,  but  regarding  commutation  are 
limited  in  overload  capacity  only  by  the  current  density  under 
the  brush. 

Such  commutating  poles  are  excited  by  series  coils,  that  is, 
coils  connected  in  series  with  the  armature  and  having  a  number 
of  effective  turns  higher  than  the  number  of  effective  series  turns 
per  armature  pole,  so  that  at  the  position  of  the  brushes  the 


FIG.   100. — Magnetic  force  distribution  with  commutating  pole. 

m.m.f.  of  the  commutating  pole  overpowers  and  reverses  the 
m.m.f.  of  the  armature,  and  produces  a  commutating  m.m.f. 
equal  to  the  product  of  the  armature  current  and  difference  of 
commutating  turns  and  armature  turns,  and  thereby  produces 
a  commutating  flux  proportional  to  the  load,  as  long  as  the  mag- 
netic flux  in  the  commutating  poles  does  not  reach  too  high 
magnetic  saturation. 

In  Fig.  100  is  shown  the  distribution  of  m.m.f.  around  the  cir- 
cumference of  the  armature,  and  in  Fig.  101  the  distribution  of 
magnetic  flux  calculated  in  the  manner  as  described  in  para- 
graphs 46  and  47.  M  represents  the  main  poles,  C  the  com- 
mutating poles.  A  main  field  excitation  FQ  is  assumed  of  10,000 
ampere-turns  per  pole,  and  an  armature  reaction  Fa  of  6000 


D.  C.  COMMUTATING  MACHINES  187 

ampere-turns  per  pole.  Choosing  then  8000  ampere-turns  per 
commutating  pole  F',  leaves  2000  ampere-turns  as  resultant  com- 
mutating  m.m.f .  at  full  load,  half  as  much  at  half  load,  etc.  The 
resultant  m.m.f.  of  the  main  field  FQ,  the  armature  Fa,  and  the 
commutating  pole  Ff  is  represented  in  Fig.  100  by  Fz,  and  the 
flux  produced  by  it  is  shown  in  Fig.  101.  As  seen,  with  the  com- 
mutator brushes  midway  between  the  field  poles,  that  is,  in  the 
center  of  the  commutating  pole,  a  commutating  flux  proportional 
to  the  armature  current  enters  the  armature  at  the  brush  B 
and  5',  and  is  cut  by  the  revolving  armature  during  commutation. 
The  use  of  the  commutating  pole  or  interpole  thus  permits 
controlling  the  commutation,  with  fixed  brush  position  midway 
between  the  field  poles,  and  commutating  poles  therefore  are 


FIG.  101. — Magnetic  flux  distribution  with  commutating  pole. 

extensively  used  in  larger  machines,  especially  of  the  high-speed 
type. 

The  commutating  pole  makes  the  commutation  independent 
of  the  main  field  strength,  and  therefore  permits  the  machines 
to  operate  with  equally  good  commutation  over  a  wide  voltage 
range,  and  at  low  voltage,  that  is,  low  field  strength,  as  required 
for  instance  in  boosters,  etc. 

50.  With  multiple-wound  armatures,  at  least  one  commutat- 
ing pole  for  every  pair  of  main  poles  is  required,  while  with  a 
series-wound  armature  a  single  commutating  pole  would  be 
sufficient  for  all  the  sets  of  armature  brushes,  if  of  sufficient 
strength.  In  general,  however,  as  many  commutating  poles  as 
main  poles  are  used. 

With  the  position  of  the  brushes  at  the  neutral,  as  is  the  case 
when  using  commutating  poles,  the  armature  reaction  has  no 


188     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

demagnetizing  component,  and  the  only  drop  of  voltage  at  load 
is  that  due  to  the  armature  resistance  drop  and  the  distortion  of 
the  main  field,  which  at  saturation  produces  a  decrease  of  the 
total  flux,  as  shown  in  Fig.  98. 

As  is  seen  in  Fig.  101,  the  magnetic  flux  of  the  commutating 
pole  is  not  symmetrical,  but  the  spread  of  flux  is  greater  at  the 
side  of  the  main  pole  of  the  same  polarity.  As  result  thereof,  the 
total  magnetic  flux  is  slightly  increased  by  the  commutating 
poles;  that  is,  the  two  halves  of  the  commutating  flux  on  the 
two  sides  of  the  brush  do  not  quite  neutralize,  and  the  com- 
mutating flux  thus  exerts  a  slight  compounding  action,  that  is, 
tends  to  raise  the  voltage.  This  can  be  still  further  increased 
by  shifting  the  brushes  slightly  back  and  thus  giving  a  magnet- 
izing component  of  armature  reaction.  This  can  be  done  with- 
out affecting  commutation  as  long  as  the  brushes  still  remain 
under  the  commutating  pole.  In  this  manner  a  compounding 
or  even  a  slight  over-compounding  can  be  produced  without  a 
series  winding  on  the  main  field  poles,  or,  inversely,  by  shifting 
the  brushes  slightly  forward,  a  demagnetizing  component  of 
armature  reaction  can  be  introduced.  Furthermore,  the  current 
induced  in  the  short-circuited  armature  coil  by  the  commutating 
field  is  magnetizing,  that  induced  by  the  magnetic  field  of  arma- 
ture reaction,  demagnetizing. 

In  operating  machines  with  commutating  poles  in  multiple, 
care  therefore  must  be  taken  not  to  have  the  compounding  action 
of  the  commutating  poles  interfere  with  the  distribution  of  load ; 
for  this  purpose  an  equalizer  connection  may  be  used  between 
the  commutating  pole  windings  of  the  different  machines,  and 
the  commutating  windings  treated  in  the  same  way  as  series 
coils  on  the  main  poles,  that  is,  equalized  between  the  different 
machines  to  insure  division  of  load. 

51.  The  advantage  of  the  commutating  pole  over  the  shift 
of  brushes  to  the  edge  of  the  next  field  pole,  in  constant  poten- 
tial machines — shunt  or  compound  wound — thus  is  that  the 
commutating  flux  of  the  former  has  the  right  intensity  at  all 
loads,  while  that  of  the  latter  is  right  only  at  one  particular  load, 
too  high  below,  too  low  above  that  load.  In  series-wound 
machines,  that  is,  machines  in  which  the  main  field  is  excited  in 
series  with  the  armature,  and  thus  varies  in  strength  with  the 
armature  current,  armature  reaction  and  field  excitation  are 
Always  proportional  to  each  other,  and  the  distribution  of  mag- 


D.  C.  COMMUTATING  MACHINES  189 

netic  flux  at  the  armature  circumference  therefore  always  has 
the  same  shape,  and  its  intensity  is  proportional  to  the  current, 
except  as  far  as  saturation  limits  it.  As  the  result  thereof, 
shifting  the  brushes  to  the  edge  of  the  field  poles,  as  in  Fig.  95, 
brings  them  in  a  field  which  is  proportional  to  the  armature  cur- 
rent and  thus  has  the  proper  intensity  as  a  commutating  field. 
Therefore  with  series-wound  machines  commutating  poles  are 
not  necessary  for  good  commutation,  but  the  shifting  of  the 
brushes  gives  the  same  result.  However,  in  cases  where  the  direc- 
tion of  rotation  frequently  reverses,  as  in  railway  motors,  the 
direction  of  the  shift  of  brushes  has  to  be  reversed  with  the  re- 
versal of  rotation.  In  railway  motors  this  cannot  be  done  with- 
out objectionable  complication,  therefore  the  brushes  have  to  be 
set  midway,  and  the  use  of  the  magnetic  flux  at  the  edge  of  the 
next  pole,  as  commutating  flux,  is  not  feasible.  In  this  case  a 
commutating  pole  is  used,  to  give,  without  mechanical  shifting 
of  the  brushes,  the  same  effect  which  a  brush  shift  would  give. 
Therefore  in  railway  motors,  especially  when  wound  for  high 
voltage,  as  1200  to  2400  volts,  a  commutating  pole  is  sometimes 
used.  This  commutating  pole,  having  a  series  winding  just 
like  the  main  pole,  changes  proportionally  with  the  main  pole. 
When  reversing  the  direction  of  rotation,  however,  the  armature 
and  the  commutating  poles  are  reversed,  while  the  main  poles 
remain  unchanged,  or  the  main  poles  are  reversed,  while  the  arma- 
ture and  the  commutating  poles  remain  unchanged;  that  is, 
the  separate  commutating  pole  becomes  necessary  because 
during  the  reversal  of  rotation  it  has  to  be  treated  differently 
from  the  main  pole. 

52.  The  commutating  pole  counteracts  the  armature  reaction 
only  at  the  place  of  commutation,  but  not  elsewhere,  and  the 
field  distribution  resulting  from  the  armature  reaction  thus  is  not 
eliminated  by  the  commutating  pole,  except  locally.  Thus  in 
machines  having  very  low  field  excitation,  and  relatively  high 
armature  reaction,  as  alternating-current  commutating  machines, 
adjustable  speed  motors  of  wide  speed  range  at  the  high-speed 
position,  boosters  near  zero  voltage,  etc.,  the  load  losses  resulting 
from  excessive  field  distortion,  the  tendency  to  instability  of 
speed,  and  the  liability  of  flashing  at  the  commutator  at  sudden 
changes  of  load  are  not  eliminated  by  the  commutating  pole, 
but  a  more  complete  neutralization  of  the  armature  reaction  is 
necessary. 


190     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


Such  is  given  by  a  compensating  winding.  This  is  a  dis- 
tributed winding,  located  in  the  field  pole  faces  closely  adjacent 
to  the  armature,  as  shown  in  Fig.  102.  It  is  connected  in  series 
but  opposition  to  the  armature  winding,  and  of  the  same  number 
of  effective  turns  as  the  armature.  By  such  a  compensating 
winding,  the  armature  reaction  is  completely  eliminated,  and  with 
it  magnetic  distortion,  load  losses,  etc. 

By  giving  the  compensating  winding  some  more  ampere-turns 
than  the  armature,  over-compensation  is  produced,  giving  a  mag- 
netic cross  flux  under  load,  opposite  to  that  of  armature  reaction, 
that  is,  a  commutating  flux.  Very  commonly  in  such  com- 
pensated machines  merely  the 
ampere-turns  of  the  compensat- 
ing winding  in  the  slots  at  the 
commutating  zone  are  increased, 
so  that  the  compensating  wind- 
ing all  around  the  armature  ex- 
actly neutralizes  the  armature 
reaction,  except  at  the  commu- 
tating zone,  where  it  over-com- 
pensates and  thus  gives  a  local 
commutating  flux.  Such  ma- 
chines, when  properly  designed, 
are  characterized  by  absence  of 
load  losses,  stability  at  all  speeds, 
instant  recovery  at  sudden  load 

changes,  and  absence  of  sparking  at  commutator  even  at  mo- 
mentary overloads  of  several  hundred  per  cent. 


FIG.  102. — Compensated  com- 
mutating machine  with  fractional 
pitch  armature  winding. 


VII.  Effect  of  Slots  on  Magnetic  Flux 

53.  With  slotted  armatures  the  pole  face  density  opposite  the 
armature  slots  is  less  than  that  opposite  the  armature  teeth,  due 
to  the  greater  distance  of  the  air  path  in  the  former  case.  Thus, 
with  the  passage  of  the  armature  slots  across  the  field  pole  a 
local  pulsation  of  the  magnetic  flux  in  the  pole  face  is  produced, 
which,  while  harmless  with  laminated  field  pole  faces,  generates 
eddy  currents  in  solid  pole  pieces.  The  frequency  of  this  pul- 
sation is  extremely  high,  and  thus  the  energy  loss  due  to  eddy 
currents  in  the 'pole  faces  may  be  considerable,  even  with  pul- 
sations of  small  amplitude.  If  S  =  peripheral  speed  of  the  arma- 


D.  C.  COMMUTATING  MACHINES 


191 


ture  in  centimeters  per  second,  lp  =  pitch  of  armature  slot  (that 
is,  width  of  one  slot  and  one  tooth  at  armature  surface),  the 

S 
frequency  is  /i  =  y-.     Or,  if  /  =  frequency   of   machine,  n  — 

number  of  armature  slots  per  pair  of  poles,  /i  =  nf. 

For  instance,/  =  33.3,  n  =  51,  thus/i  =  1700. 

Under  the  assumption,  width 
of  slots  equals  width  of  teeth 
=  2  X  width  of  air  gap,  the  dis- 
tribution of  magnetic  flux  at  the 
pole  face  is  plotted  in  Fig.  103. 

The  drop  of  density  opposite 
each  slot  consists  of  two  curved 
branches  equal  to  those  in  Fig. 
92,  that  is,  calculated  by 


•B' 


-3 

n 

FIG.  103.—  I 

< 

« 

i 

slots  on  flu 

Iffect  of 

B 


distribution. 


V  + 1*2 

The  average  flux  is  7525;  that  is,  by  cutting  half  the  armature 
surface  away  by  slots  of  a  width  equal  to  twice  the  length  of  air 
gap,  the  total  flux  under  the  field  pole  is  reduced  only  in  the 
proportion  8000  to  7525,  or  about  6  per  cent. 

The  flux  B  pulsating  between  8000  and  5700  is  equivalent  to 
a  uniform  flux  B\  =  7525  superposed  with  an  alternating  flux 


FIG.  104. — Effect  of  slots  on  flux  distribution. 

BO,  shown  in  Fig.  104,  with  a  maximum  of  475  and  a  minimum 
of  1825.  This  alternating  flux  BQ  can,  as  regards  production  of 
eddy  currents,  be  replaced  by  the  equivalent  sine  wave  B0o,  that 
is,  a  sine  wave  having  the  same  effective  value  (or  square  root  of 
mean  square).  The  effective  value  is  718. 

The  pulsation  of  magnetic  flux  farther  in  the  interior  of  the 
field-pole  face  can  be  approximated  by  drawing  curves  equi- 


192     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

distant  from  BQ.  Thus  the  curves  #0.5,  BI>  ^1.5,  #2,  #2.5,  and  B3 
are  drawn  equidistant  from  B0  in  the  relative  distances  0.5,  1, 
1.5,  2,  2.5,  and  3  (where  la  =  1  is  the  length  of  air  gap).  They 
give  the  effective  values: 

BQ  BQ.S  BI  BI.Z  BZ  Bz.s  B3 

718          373  184  119  91  69  57 

That  is,  the  pulsation  of  magnetic  flux  rapidly  disappears 
toward  the  interior  of  the  magnet  pole,  and  still  more  rapidly 
the  energy  loss  by  eddy  currents,  which  is  proportional  to  the 
square  of  the  magnetic  density. 

54.  In  calculating  the  effect  of  eddy  currents,  the  magnetizing 
effect  of  eddy  currents  may  be  neglected  (which  tends  to  reduce 
the  pulsation  of  magnetism);  this  gives  the  upper  limit  of  loss 

Let  B  =  effective  density  of  the  alternating  magnetic  flux, 
S  =  peripheral   speed    of    armature    in    centimeters    per 

second,  and 
I  =  length  of  pole  face  along  armature. 

The  e.m.f.  generated  in  the  pole  face  is  then 

e  =  SIB  X  10-8, 
and  the  current  in  a  strip  of  thickness  Al  and  1  cm.  width, 

eAl      SIBAl  10~8       SBAl  10~8 

Ai  =  —r  =  -     — jr—    -  =  -  —i 

pl>  pi  p 

where 

p  =  resistivity  of  the  material. 

Thus  the  effect  of  eddy  currents  in  this  strip  is 

'  SHB2Al  10-16 

Ap  =  eAi  =  -  —i 

or  per  cubic  centimeter, 

S*B*  10~16 

P  =        ~~P~ 

that  is,  proportional  to  the  square  of  the  effective  value  of  mag- 
netic pulsation,  the  square  of  peripheral  speed,  and  inversely 
proportional  to  the  resistivity. 
Thus,  assuming  for  instance, 
S  =  2000, 

p  =  20  X  10~6,  for  cast  steel, 
p  =  100  X  10~6,  for  cast  iron, 
we  have  in  the  above  example, 


D.  C.  COMMUTATING  MACHINES 


193 


At  distance 
from  pole 
face 

B 

p 

Cast  steel 

Cast  iron 

0 

718 

10.3                  2.06 

la 

2 

373 

2.78 

0.56 

la 

184 

0.677 

0.135 

~2~ 

119 

0.283 

0.057 

2  la 

91 

0.166 

0.033 

5  la 
2 

69 

0.095 

0.019 

3  la 

57 

0.065 

0.013 

VIII.  Armature  Reaction 

55.  At  no  load,  that  is,  with  no  current  in  the  armature  cir- 
cuit, the  magnetic  field  of  the  commutating  machine  is  sym- 
metrical with  regard  to  the  field  poles. 

Thus  the  density  at  the  armature  surface  is  zero  at  the  point 
or  in  the  range  midway  between  adjacent  field  poles.  This 
point,  or  range,  is  called  the  "neutral"  point  or  "neutral"  range 
of  the  commutating  machine. 

Under  load  the  armature  current  represents  a  m.m.f.  acting  in 
the  direction  from  commutator  brush  to  commutator  brush  of 
opposite  polarity,  that  is,  in  quadrature  with  the  field  m.m.f.  if 
the  brushes  stand  midway  between  the  field  poles;  or  shifted 
against  the  quadrature  position  by  the  same  angle  by  which  the 
commutator  brushes  are  shifted,  which  angle  is  called  the  angle 
of  lead. 

If  n  =  turns  in  series  between  brushes  per  pole,  and  i  =  cur- 
rent per  turn,  the  m.m.f.  of  the  armature  is  Fa  =  ni  per  pole. 
Or,  if  r?o  =  total  number  of  turns  on  the  armature,  nc  =  number 
of  turns  or  circuits  in  multiple,  2np  =  number  of  poles,  and  t'0 
=  total  armature  current,  the  m.m.f.  of  the  armature  per  pole  is 

Fa  =  ^  —     This  m.m.f.  is  called  the  armature  reaction  of  the 
2npnc 

continuous-current  machine. 

Since  the  armature  turns  are  distributed  over  the  total  pitch 
of  pole,  that  is,  a  space  of  the  armature  surface  representing 
180  deg.,  the  resultant  armature  reaction  is  found  by  multiplying 


194     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

C    -j-  go        2 
Fa  with  the  average  cos  =  — ,  and  is  thus 

—  yu      ^" 


Fao 


2  Fa       2  ni 


When  comparing  the  armature  reaction  of  commutating  ma- 
chines with  other  types  of  machines,  as  synchronous  machines 

2  Fa 

etc.,  the  resultant  armature  reaction  Fao  =  -    -  has  to  be  used. 

In  discussing  commutating  machines  proper,  however,  the  value 
Fa  =  ni  is  usually  considered  as  the  armature  reaction. 

56.  The  armature  reaction  of  the  commutating  machine  has  a 
distorting  and  a  magnetizing  or  demagnetizing  action  upon  the 
magnetic  field.     The  armature  ampere-turns  beneath  the  field 
poles  have  a  distorting  action  as  discussed  under  "  Magnetic  Dis- 
tribution" in  the  preceding  paragraphs.     The  armature  ampere- 
turns  between  the  field  poles  have  no  effect  upon  the  resultant 
field  if  the  brushes  stand  at  the  neutral;  but  if  the  brushes  are 
shifted,  the  armature  ampere-turns  inclosed  by  twice  the  angle 
of  lead  of  the  brushes  have  a  demagnetizing  action. 

Thus,  if  r  =  pole  arc  as  fraction  of  pole  pitch,  TI  =  shift  of 
brushes  as  fraction  of  pole  pitch,  Fa  the  m.m.f.  of  armature 
reaction,  and  FQ  the  m.m.f.  of  field  excitation  per  pole,  the  demag- 
netizing component  of  armature  reaction  is  riFa,  the  distorting 
component  of  armature  reaction  is  rFa,  and  the  magnetic  density 
at  the  strengthened  pole  corner  thus  corresponds  to  the  m.m.f. 

rFa                                                                                                     rFa 
FQ  +  -£-  at  the  weakened  field  corner  to  the  m.m.f.  FQ g— 

IX.  Saturation  Curves 

57.  As  saturation  curve  or  magnetic  characteristic  of  the  com- 
mutating machine  is  understood  the  curve  giving  the  generated 
voltage,  or  terminal  voltage  at  open  circuit  and  normal  speed, 
as  function  of  the  ampere-turns  per  pole  field  excitation. 

Such  curves  are  of  the  shape  shown  in  Fig.  105  as  A.  Owing 
to  the  remanent  magnetism  or  hysteresis  of  the  iron  part  of  the 
magnetic  circuit,  the  saturation  curve  taken  with  decreasing 
field  excitation  usually  does  not  coincide  with  that  taken  with 
increasing  field  excitation,  but  is  higher,  and  by  gradually  first 
increasing  the  field  excitation  from  zero  to  maximum  and  then 
decreasing  again,  the  looped  curve  in  Fig.  106  is  derived,  giving 


D.  C.  COMMUTATING  MACHINES 


195 


as  average  saturation  curve  the  curve  shown  in  Fig.  105  as  A 
and  as  central  curve  in  Fig.  106. 

Direct-current  generators  are  usually  operated  at  a  point  of 
the  saturation  curve  above  the  bend,  that  is,  at  a  point  where  the 
terminal  voltage  increases  considerably  less  than  proportionally 
to  the  field  excitation.  This  is  necessary  in  self-exciting  direct- 
current  generators  to  secure  stability. 

The  ratio 


increase  of  field  excitation 
total  field  excitation 

that  is, 


corresponding  increase  of  voltage 

total  voltage 
F*      de 


FIG.  105. — Saturation  characteristics. 

is  called  saturation  factor  s,  and  is  plotted  in  Fig.  105.  It  is  the 
ratio  of  a  small  percentage  increase  in  field  excitation  to  a  corre- 
sponding percentage  increase  in  voltage  thereby  produced.  The 

quantity  1 is  called  the  percentage  saturation  of  the  ma- 

s 

chine,  as  it  shows  the  approach  of  the  machine  field  to  mag- 
netic saturation. 

58.  Of  considerable  importance  also  are  curves  giving  the 
terminal  voltage  as  function  of  the  field  excitation  at  load. 
Such  curves  are  called  load  saturation  curves,  and  can  be  constant 


196     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

current  load  saturation  curve,  that  is,  terminal  voltage  as  func- 
tion of  field  ampere-turns  at  constant  full-load  current  through 
the  armature,  and  constant  resistance  load  saturation  curve,  that 
is,  terminal  voltage  as  function  of  field  ampere-turns  if  the 
machine  circuit  is  closed  through  a  constant  resistance  giving 
full-load  current  at  full-load  terminal  voltage. 

A  constant  current  load  saturation  curve  is  shown  as  B,  and 
a  constant  resistance  load  saturation  curve  as  C  in  Fig.  105. 


FIG.  106. — Saturation  curves. 

X.  Compounding 

59.  In  the  direct-current  generator  the  field  excitation  re- 
quired to  maintain  constant  terminal  voltage  has  to  be  increased 
with  the  load.     A  curve  giving  the  field  excitation  in  ampere- 
turns  per  pole,  as  function  of  the  load  in  amperes,  at  constant 
terminal  voltage,  is  called  the  compounding  curve  of  the  machine. 
The  increase  of  field  excitation  required  with  load  is  due  to : 
1.  The  internal  resistance  of  the  machine,  which  consumes 
e.m.f.  proportional  to  the  current,  so  that  the  generated  e.m.f., 
and  thus  the  field  m.m.f.  corresponding  thereto,  has  to  be  greater 
under  load.     If  p  =  resistance  drop  in  the  machine  as  fraction 

ir 
of  terminal  voltage,  =  —  >    the  generated  e.m.f.  at  load  has  to  be 

£ 

e  (1  +  p),  and  if  ^o=  no-load   field   excitation,  and  s  =  satu- 


D.  C.  COMMUTATING  MACHINES  197 

ration  coefficient,  the  field  excitation  required  to  produce  the 
e.m.f .  e  (1  +  p)  is  Fo  (1  +  sp) ;  thus  an  additional  excitation  of 
spF0  is  required  at  load,  due  to  the  armature  resistance. 

2.  The  demagnetizing  effect  of  the  ampere-turns  armature 
reaction  of  the  angle  of  shift  of  brushes  TI  requires  an  increase 
of  field  excitation  by  riFa.     (Section  VII.) 

3.  The  distorting  effect  of  armature  reaction  does  not  change 
the  total  m.m.f.  producing  the  magnetic  flux.     If,  however,  mag- 
netic saturation  is  reached  or  approached  in  a  part  of  the  mag- 
netic circuit  adjoining  the  air  gap,  the  increase  of  magnetic 
density  at  the  strengthened  pole  corner  is  less  than  the  decrease 
at  the  weakened  pole  corner,  and  thus  the  total  magnetic  flux 
with  the  same  total  m.m.f.  is  reduced,  and  to  produce  the  same 
total  magnetic  flux  an  increased  total  m.m.f.,  that  is,  increase  of 
field  excitation,  is  required.     This  increase  depends  upon  the 
saturation  of  the  magnetic  circuit  adjacent  to  the  armature 
conductors. 

4.  The  magnetic  stray  field  of  the  machine,  that  is,  that  part 
of  the  magnetic  flux  which  passes  from  field  pole  to  field  pole  with- 
out entering  the  armature,  usually  increases  with  the  load.     This 
stray  field  is  proportional  to  the  difference  of  magnetic  potential 
between  field  poles;  that  is,  at  no  load  it  is  proportional  to  the 
ampere-turns  m.m.f.  consumed  in  air  gap,  armature  teeth,  and 
armature  core.     Under  load,  with  the  same  generated  e.m.f., 
that  is,  the  same  magnetic  flux  passing  through  the  armature 
core,  the  difference  of  magnetic  potential  between  adjacent  field 
poles  is  increased  by  the  counter  m.m.f.  of  the  armature  and  by 
saturation.     Since  this  magnetic  stray  flux  passes  through  field 
poles  and  yoke,  the  magnetic  density  therein  is  increased  and  the 
field  excitation  correspondingly,  especially  if  the  magnetic  den- 
sity in  field  poles  and  yoke  is  near  saturation.     This  increase  of 
field  strength  required  by  the  increase  of  density  in  the  external 
magnetic  circuit,  due  to  the  increase  of  magnetic  stray  field, 
depends  upon  the  shape  of  the  magnetic  circuit,  the  armature 
reaction,  and  the  saturation  of  the  external  magnetic  circuit. 

Curves  giving,  with  the  amperes  output  as  abscissas,  the 
ampere-turns  per  pole  field  excitation  required  to  increase  the 
voltage  proportionally  to  the  current  are  called  over-compounding 
curves.  In  the  increase  of  field  excitation  required  for  over- 
compounding,  the  effects  of  magnetic  saturation  are  still  more 
marked. 

13 


198     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

XL  Characteristic  Curves 

60.  The  field  characteristic  or  regulation  curve,  that  is,  curve 
giving  the  terminal  voltage  as  function  of  the  current  output  at 
constant  field  excitation,  is  of  less  importance  in  commutating 
machines  than  in  synchronous  machines,   since   commutating 
machines  are  usually  not  operated  with  separate  and  constant 
excitation,  and  the  use  of  the  series  field  affords  a  convenient 
means  of  changing  the  field  excitation  proportionally  to  the  load. 
The  curve  giving  the  terminal  voltage  as  function  of  current  out- 
put, in  a  compound-wound  machine,  at  constant  resistance  in  the 
shunt  field,  and  constant  adjustment  of  the  series  field,  is,  how- 
ever, of  importance  as  regulation  curve  of  the  direct-current 
generator.     This  curve  would  be  a  straight  line  except  for  the 
effect  of  saturation,  etc.,  as  discussed  above. 

XII.  Efficiency  and  Losses 

61.  The  losses  in  a  commutating  machine  which  have  to  be 
considered  when  deriving  the  efficiency  by  adding  the  individual 
losses  are: 

1.  Loss  in  the  resistance  of  the  armature,  the  commutator 
leads,  brush  contacts  and  brushes,  in  the  shunt  field  and  the  series 
field  with  their  rheostats. 

2.  Hysteresis  and  eddy  currents  in  the  iron  at  a  voltage  equal 
to  the  terminal  voltage,  plus  resistance  drop  in  a  generator,  or 
minus  resistance  drop  in  a  motor. 

3.  Eddy  currents  in  the  armature  conductors  when  large  and 
not  protected,  and  in  pole  faces  when  solid  and  the  air  gap 
is  small. 

4.  Friction  of  bearings,  of  brushes  on  the  commutator,  and 
windage. 

5.  Load  losses,  due  to  the  increase  of  hysteresis  and  of  eddy 
currents  under  load,  caused  by  the  change  of  the  magnetic  dis- 
tribution, as  local  increase  of  magnetic  density  and  of  stray  field. 

The  friction  of  the  brushes  and  the  loss  in  the  contact  resist- 
ance of  the  brushes  are  frequently  quite  considerable,  especially 
with  low-voltage  machines. 

Constant  or  approximately  constant  losses  are:  friction  of 
bearings  and  of  commutator  brushes,  and  windage;  hysteresis 
and  eddy  current  losses;  and  shunt  field  excitation.  Losses 


D.'C.  COMMUTATING  MACHINES  199 

increasing  with  the  load,  and  proportional  or  approximately 
proportional  to  the  square  of  the  current:  armature  resistance 
losses;  series  field  resistance  losses;  brush  contact  resistance 
losses;  and  the  so-called  "load  losses." 

XIII.  Commutation 

62.  The  most  important  problem  connected  with  commutating 
machines  is  that  of  commutation. 

Fig.  107  represents  diagrammatically  a  commutating  machine. 


FIG.  107. — Diagram  for  the  study  of  commutation. 

The  e.m.f.  generated  in  an  armature  coil  A  is  zero  with  this  coil 
at  or  near  the  position  of  the  commutator  brush  B\.  It  rises 
and  reaches  a  maximum  about  midway  between  two  adjacent 
sets  of  brushes,  BI  and  B2,  at  C,  and  then  decreases  again, 
reaching  zero  at  or  about  B2,  and  then  repeats  the  same  change 
in  opposite  direction.  The  current  in  armature  coil  A,  however, 
is  constant  during  the  motion  of  the  coil  from  BI  to  BI.  While 
the  coil  A  passes  the  brush  B2,  however,  the  current  in  the  coil 
A  reverses,  and  then  remains  constant  again  in  opposite  direc- 


200     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

tion  during  the  motion  from  B2  to  B3.  Thus,  while  the  armature 
coils  of  a  commutating  machine  are  the  seat  of  a  system  of  poly- 
phase e.m.fs.  having  as  many  phases  as  coils,  the  current  in  these 
coils  is  constant,  reversing  successively. 

63.  The  reversal  of  current  in  coil  A  takes  place  while  the 
gap  G  between  the  two  adjacent  commutator  segments  between 
which  the  coil  A  is  connected  passes  the  brush  B2.     Thus,  if 
lw  =  width  of  brushes,  S  =  peripheral  speed  of  commutator  per 
second  in  the  same  measure  in  which  lw  is  given,  as  in  inches  per 

second  if  Z»  is  given  in  inches,  to  =  -£  is  the  time  during  which 

the  current  in  A  reverses.     Thus,  considering  the  reversal  as  a 

1  S 

single  alternation,  tQ  is  a  half  period,  and  thus  /0  =  ^-7-  =  ;ry-  is 

4  »o      z  iw 

the  frequency  of  commutation;  hence,  if  L  =  inductance  of  the 
armature  coil  A,  the  e.m.f.  generated  in  the  armature  coil  during 
commutation  is  eo  =  2irfoLiot  where  io  =  current  reversed,  and 
the  energy  which  has  to  be  dissipated  during  commutation  is  i<pL.~ 
The  frequency  of  commutation  is  very  much  higher  than  the 
frequency  of  synchronous  machines,  and  averages  from  300  to 
1000  cycles  per  second,  or  more. 

64.  In  reality,  however,  the  changes  of  current  during  com- 
mutation are  not  sinusoidal,  but  a  complex  exponential  func- 
tion, and  the  resistance  of  the  commutated  circuit   enters  the 
problem  as  an  important  factor.     In  the  moment  when  the  gap 
G  of  the  armature  coil  A  reaches  the  brush  Bz,  the  coil  A  is  short- 
circuited  by  the  brush,  and  the  current  iQ  in  the  coil  begins  to 
die  out,  or  rather  to  change  at  a  rate  depending  upon  the  internal 
resistance  and  the  inductance  of  the  coil  A  and  the  e.m.f.  gener- 
ated in  the  coil  by  the  magnetic  flux  of  armature  reaction  and  by 
the  field  magnetic  flux.     The  higher  the  internal  resistance  the 
faster  is  the  change  of  current,  and  the  higher  the  inductance 
the  slower  the  current  changes.     Thus  two  cases  have  to  be  dis- 
tinguished. 

1.  No  magnetic  flux  enters  the  armature  at  the  position  of 
the  brushes,  that  is,  no  e.m.f.  is  generated  in  the  armature  coil 
under  commutation,  except  that  of  its  own  self-inductance.     In 
this  case  the  commutation  is  entirely  determined  by  the  induc- 
tance and  resistance  of  the  armature  coil  A,  and  is  called  re- 
sistance commutation. 

2.  Commutation  takes  place  in  an  active  magnetic  field;  that 


D.  C.  COMMUTATING  MACHINES  201 

is,  in  the  armature  during  commutation  an  e.m.f.  is  generated 
by  its  rotation  through  a  magnetic  field.  This  magnetic  field 
may  be  the  magnetic  field  of  armature  reaction,  or  the  reverse 
magnetic  field  of  a  commutating  pole,  or  the  fringe  of  the  main 
field  of  the  machine,  into  which  the  brushes  are  shifted.  In 
this  case  the  commutation  depends  upon  the  inductance  and  the 
resistance  of  the  armature  coil  and  the  e.m.f.  generated  therein 
by  the  main  magnetic  field,  and  if  this  magnetic  field  is  a  corn- 
mutating  field,  is  called  voltage  commutation. 

In  either  case  the  resistance  of  the  brushes  and  their  contact 
may  either  be  negligible,  as  usually  the  case  with  copper  brushes, 
or  it  may  be  of  the  same  or  a  higher  magnitude  than  the  internal 
resistance  of  the  armature  coil  A.  The  latter  is  usually  the  case 
with  carbon  or  graphite  brushes. 

In  the  former  case  the  resistance  of  the  short-circuit  of  arma- 
ture coil  A  under  commutation  is  approximately  constant;  in 
the  latter  case  it  varies  from  infinity  in  the  moment  of  beginning 
commutation  down  to  minimum,  and  then  up  again  to  infinity  at 
the  end  of  commutation. 

65.  (a)  Negligible  resistance  of  brush  and  brush  contact. 

This  is  more  or  less  approximately  the  case  with  copper  brushes. 

Let  iQ  =  current, 

L  =  inductance, 

r  —  resistance  of  armature  coil, 

to  =  -£•  =  time  of  commutation, 

and  —  e  =  e.m.f.  generated  in  the  armature  coil  by  its  rotation 
through  the  magnetic  field,  where  e  is  negative  for  the  magnetic 
field  of  armature  reaction  and  positive  for  the  commutating  field. 
Denoting  the  current  in  the  coil  A  at  time  t  after  beginning 
of  commutation  by  i,  the  e.m.f.  of  self-inductance  is 

_  di 


Thus  the  total  e.m.f.  acting  in  coil  A, 

di 

—  e  -\-  ei  =  —  e  —  L  -77* 
at 

and  the  current  is 

e      Ldi 
r      r  dt 


202     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Transposing,  this  expression  becomes 

rdt         di 


the  integral  of  which  is 

-  j-  =  loge  (^  +  i)  -  loge  c, 


where  loge  c  =  integration  constant. 
Since  at  t  =  0,  i  =  io,  we  have 


loge  c  =  log       +  i0  , 
therefore 

(g  \  In 

-  +  i,J  ,  and  ^  =  (-  + 

and,  at  the  end  of  commutation,  or,  t  =  to, 


For  perfect  commutation, 

that  is,  the  current  at  the  end  of  commutation  must  have  reversed 
and  reached  its  full  value  in  opposite  direction. 

Substituting  in  this  last  equation  the  value  i\  from  the  pre- 
ceding equation,  and  transforming,  we  have 


taking  the  logarithms  of  both  terms, 


L   o 

-t-> 

and,  solving  the  exponential  equation  for  e,  we  obtain 


D.  C.  COMMUTATING  MACHINES 


203 


It  is  evident  that  the  inequality  e  >  i<>r  must  be  true,  otherwise 

perfect  commutation  is  not  possible. 

If 


we  have 


that  is,  the  current  never  reverses,  but  merely  dies  out  more  or 
less,  and  in  the  moment  when  the  gap  G  of  the  armature  coil 
leaves  the  brush  B  the  current  therein  has  to  rise  suddenly  to 
full  intensity  in  opposite  direction.  This  being  impossible,  due 
to  the  inductance  of  the  coil,  the  current  forms  an  arc  from  the 
brush  across  the  commutator  surface  for  a  length  of  time  depend- 
ing upon  the  inductance  of  the  armature  coil. 

Therefore,  with  low-resistance  brushes,  resistance  commutation 
is  not  permissible  except  with  machines  of  extremely  low  arma- 


FIG.  108. — Brush  commutating  coil  A. 

ture  inductance,  that  is,  armature  inductance  so  low  that  the 

magnetic  energy  -7^—,  which  appears  as  "spark"  in  this  case,  is 
& 

harmless. 

Voltage  commutation  is  feasible  with  low-resistance  brushes, 
but  requires  a  commutating  e.m.f.  e  proportional  to  current  z'o; 
that  is,  requires  shifting  of  brushes  proportionally  to  the  load,  or 
a  commutating  pole. 

In  the  preceding,  the  e.m.f.  e.has  been  assumed  constant  dur- 
ing the  commutation.  In  reality  it  varies  somewhat,  usually 
increasing  with  the  approach  of  the  commutated  coil  to  a  denser 
field.  It  is  not  possible  to  consider  this  variation  in  general,  and 
e  is  thus  to  be  considered  the  average  value  during  commutation. 

66.  (b)  High-resistance  brush  contact. 

Fig.  108  represents  a  brush  B  commutating  armature  coil  A. 


204     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Let  r0  =  contact  resistance  of  the  brush,  that  is,  resistance 
from  the  brush  to  the  commutator  surface  over  the  total  bearing 
surface  of  the  brushes.  The  resistance  of  the  commutated  cir- 
cuit is  thus  internal  resistance  of  the  armature  coil  r  plus  the 
resistance  from  C  to  B  plus  the  resistance  from  B  to  D. 

Thus,  if  to  =  time  of  commutation,  at  the  time  t  after  the  be- 
ginning of  the  commutation,  the  resistance  from  C  to  B  is  — 

and  from  B  to  D  is -;  thus,  the  total  resistance  of  the  corn- 
to  —  * 

mutated  coil  is 

T-»  ,      to?*0     i         'o7*0  to  TQ 

R  =  r  +  —  +  -. :  =  r  +     (          v 

t         to  —  t  t  (to  —  t) 

If  i0  =  current  in  coil  A  before  commutation,  the  total  cur- 
rent into  the  armature  from  brush  B  is  2  i0.  Thus,  if  i  —  current 
in  commutated  coil,  the  current  from  B  to  D  =  iQ  +  i,  the  cur- 
rent from  B  to  C  =  io  —  i. 

Hence,  the  difference  of  potential  from  D  to  C  is 


The  e.m.f.  acting  in  coil  A  is 

Ldi 


and  herefrom  the  difference  of  potential  from  D  to  C  is 

L--  >• 

hence, 

di        .  tofo     , .         .>.        tofo  / .         .\ 

dt  to  —  t  t 

or,  transposing, 

Ldi  .        toToio  (2 1  —  to)          tozroi 

dt  t  (to  —  t)  t  (to  —  t) 

T  di  .  /  roto2     \        Totoio  C2>t  —  to) 


'    t(tQ-t)  * 

The  further  solution  of  this  general  problem  becomes  difficult, 
but  even  without  integrating  this  differential  equation  a  number 
of  important  conclusions  can  be  derived. 

Obviously  the  commutation  is  correct  and  thus  sparkless,  if 


D.  C.  COMMUTATING  MACHINES  205 

the  current  entering  over  the  brush  shifts  from  segment  to  seg- 
ment in  direct  proportion  to  the  motion  of  the  gap  between  ad- 
jacent segments  across  the  brush,  that  is,  if  the  current  density 
is  uniform  all  over  the  contact  surface  of  the  brush.  This  means 
that  the  current  i  in  the  short-circuited  coil  varies  from  +  io  to 
—  iQ  as  a  linear  function  of  the  time.  In  this  case  it  can  be  rep- 
resented by 

.       .  to-2t 

^  =  ^o — r— J 
to 

thus, 

di  =       2ip 
dt~         to  * 

Substituting  this  value  in  the  general  differential  equation 
gives,  after  some  transformation, 

-?-*<>  +  r(to  -  20  -  2L  =  0; 
or, 

e  =  i    I  — 

which  gives  at  the  beginning  of  commutation,  t  =  0, 


at  the  end  of  commutation,  t  =  tQ, 


that  is,  even  with  high-resistance  brushes,  for  perfect  com- 
mutation, voltage  commutation  is  necessary,  and  the  e.m.f.  e 
impressed  upon  the  commutated  coil  must  increase  during  com- 
mutation from  ei  to  62,  by  the  above  equation.  This  e.m.f.  is 
proportional  to  the  current  iQ>  but  is  independent  of  the  brush 
resistance  r0. 

RESISTANCE  COMMUTATION 

67.  Herefrom  it  follows  that  resistance  commutation  cannot 
be  perfect,  but  that  at  the  contact  with  the  segment  that  leaves 
the  brush  the  current  density  must  be  higher  than  the  average. 
Let  g  =  ratio  of  actual  current  density  at  the  moment  of  leaving 
the  brush  to  average  current  density  of  brush  contact,  and  con- 


20G     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

sidering  only  the  end  of  commutation,  as  the  most  important 
moment,  we  have 

.       .  (2g  -l)U-2gt 

I     —    lQ  --  —  -  --    • 

to 

For  t  =  to  —  tl  this  gives 

^1 

i  =  -  io  +  2  g  -  io, 
to 

while  uniform  current  density  would  require 

^1 

i  =  —  io  +  2  -  i0. 
to 

The  general  differential  equation  of  resistance  commutation, 
e  =  0,  is 

di  rplo2    \      rQt0i0(2t-  t0) 


Substituting  in  this  equation  the  value  of  i  from  the  foregoing 
equation,  expanding  and  cancelling  to  —  t,  we  obtain 

2  r0*o2  (g  -  1)  +.  rtt0  (2  g  -  1)  -  2  grt2  -  2  gLt  =  0; 
hence, 

f  rt) 


g  ~  2  (r0*02  +  rtto  -  rt*  -  Lt) 
and  for  t  =  t0, 


2(rQtQ-L)  ~rQtQ-L' 

that  is,  g  is  always  >  1. 

The  smaller  L  and  the  larger  rQ)  the  smaller  is  g;  that  is,  the 
nearer  it  is  to  1,  the  condition  of  perfect  commutation,  and 
the  better  is  the  commutation. 

Sparkless  commutation  is  impossible  for  very  large  values  of 
g,  that  is,  when  L  approaches  roto,  or  when  r0  is  not  much  larger 

than  —  •     For  this  reason,  in  machines  in  which  L  cannot  be 
£o 

made  small,  r  is  sometimes  made  large  by  inserting  resistors 
in  the  leads  between  the  armature  and  the  commutator,  so-called 
''resistance"  or  "preventive"  leads  as  used  in  alternating-current 
commutator  motors. 

XIV.  Types  of  Commutating  Machines 

68.  By  the  methods  of  excitation,  commutating  machines 
are  subdivided  into  magneto,  separately  excited,  shunt,  series, 


D.  C.  COMMUTATING  MACHINES 


207 


and  compound  machines.  Magneto  machines  and  separately 
excited  machines  are  very  similar  in  their  characteristics.  In 
either,  the  field  excitation  is  of  constant,  or  approximately 
constant,  impressed  m.m.f.  Magneto  machines,  however,  are 
little  used,  except  for  very  small  sizes. 

By  the  direction  of  energy  transformation,  commutating  ma- 
chines are  subdivided  into  generators  and  motors. 

Of  foremost  importance  in  discussing  the  different  types  of 
machines  is  the  saturation  curve  or  magnetic  characteristic; 
that  is,  a  curve  relating  terminal  voltage  at  constant  speed  to 
ampere-turns  per  pole  field  excitation,  at  open  circuit.  Such 
a  curve  is  shown  as  A  in  Figs.  109  and  110.  It  has  the  same 


1      8      9      4      5      6       78      9     19    U     12    13     H    15    16    I?     J«     W    20     81 

FIG.  109. — Generator  saturation  curves. 


general  shape  as  the  magnetic  flux  density  curve,  except  that  the 
knee  or  bend  is  less  sharp,  due  to  the  different  parts  of  the 
magnetic  circuit  saturation  successively. 

Thus,  in  order  to  generate  voltage  ac  the  field  excitation  oc 
is  required.  Subtracting  from  ac  in  a  generator,  Fig.  109,  or 
adding  in  a  motor,  Fig.  110,  the  value  ab  =  ir,  the  voltage  con- 
sumed by  the  resistance  of  the  armature,  commutator,  etc., 
gives  the  terminal  voltage  be  at  current  i,  and  adding  to  oc  the 
value  ce  =  bd  =  iq  =  armature  reaction,  or  rather  field  excita- 
tion required  to  overcome  the  armature  reaction,  gives  the  field 
excitation  oe  required  to  produce  the  terminal  voltage  de  at 


208     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

current  i.  The  armature  reaction  iq,  corresponding  to  current  i, 
is  calculated  as  discussed  before,  and  q  may  be  called  the  coef- 
ficient of  armature  reaction. 

69.  Such  a  curve,  D,  shown  in  Fig.  109  for  a  generator,  and 
in  Fig.  110  for  a  motor,  and  giving  the  terminal  voltage  de  at 
current  i,  corresponding  to  the  field  excitation  oe}  is  called  a 
load  saturation  curve.  Its  points  are  respectively  distant  from 
the  corresponding  points  of  the  no-load  saturation  curve  A  a 
constant  distance  equal  to  ad,  measured  parallel  thereto. 

Curves  D  are  plotted  under  the  assumption  that  the  armature 
reaction  is  constant.  Frequently,  however,  at  lower  voltage  the 


1231567 

FIG.  110. 


9   10  11  12  13  14  .15  16  17  18  19  20  21 

-Motor  saturation  curves. 


armature  reaction,  or  rather  the  increase  of  excitation  required 
to  overcome  the  armature  reaction  iq,  increases,  since  with 
voltage  commutation  at  lower  voltage,  and  thus  weaker  field 
strength,  the  brushes  have  to  be  shifted  more  to  secure  spark- 
less  commutation,  and  thus  the  demagnetizing  effect  of  the 
angle  of  lead  increases.  At  higher  voltage  iq  usually  increases 
also,  due  to  increase  of  magnetic  saturation  under  load,  caused 
by  the  increased  stray  field.  Thus,  the  load  saturation  curve 
of  the  continuous-current  generator  more  or  less  deviates  from 
the  theoretical  shape  D  toward  a  shape  shown  as  G. 


D.  C.  COMMUTATING  MACHINES 
A.  GENERATORS 


209 


Separately  Excited  and  Magneto  Generator 

70.  In  a  separately  excited  or  magneto  machine,  that  is,  a 
machine  with   constant  field  excitation  FQ)   a  demagnetization 


\ 


\\ 


10  20  30  40   50  60  70  80  90  100  110  120  130  140  150  160 

FIG.  111. — Separately  excited  or  magneto-generator  demagnetization  curve 
and  load  characteristic  with  constant  shift  of  brushes. 


10   20  30  40  50  60  70  80   90  100  110  120  130 

FIG.  112. — Separately  excited  or  magneto-generator  demagnetization  curve 
and  load  characteristic  with  variable  shift  of  brushes. 

curve  can  be  plotted  from  the  magnetization  or  saturation  curve 
A  in  Fig.  109.  At  current  i,  the  resultant  m.m.f .  of  the  machine 
is  FQ  —  iq,  and  the  generated  voltage  corresponds  thereto  by 
the  saturation  curve  A  in  Fig.  110.  Thus,  in  Fig.  Ill  a  de- 
magnetization curve  A  is  plotted  with  the  current  ob  =  i  as 


210     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

abscissas  and  the  generated  e.m.f.  ab  as  ordinates,  under  the 
assumption  of  constant  coefficient  of  armature  reaction  q,  that 
is,  corresponding  to  curve  D  in  Fig.  109.  This  curve  becomes 
zero  at  the  current  ?o,  which  makes  i$q  =  FQ.  Subtracting  from 
curve  A  in  Fig.  Ill  the  drop  of  voltage  in  the  armature  and 
commutator  resistance,  ac  =  ir,  gives  the  external  characteristic 
B  of  the  machine  as  generator,  or  the  curve  relating  the  terminal 
voltage  to  the  current. 

In  Fig.  112  the  same  curves  are  shown  under  the  assumption 
that  the  armature  reaction  varies  with  the  voltage  in  the  way 
as  represented  by  curve  G  in  Fig.  109. 

In  a  separately  excited  or  magneto  motor  at  constant  speed 
the  external  characteristic  would  lie  as  much  above  the  demag- 
netization curve  A  as  it  lies  below  in  a  generator  in  Fig.  Ill, 
and  at  constant  voltage  the  speed  would  vary  inversely  pro- 
portional hereto. 

Shunt  Generator 

71.  The  external  or  load  characteristic  of  the  shunt  generator 
is  plotted  in  Fig.  113  with  the  current  as  abscissas  and  the 
terminal  voltage  as  ordinates,  as  A  for  constant  coefficient  of 
armature  reaction,  and  as  B  for  a  coefficient  of  armature  reac- 
tion varying  with  the  voltage  in  the  way  as  shown  in  G,  Fig.  109. 
The  construction  of  these  curves  is  as  follows: 

In  Fig.  109,  og  is  the  straight  line  giving  the  field  excitation 
oh  as  function  of  the  terminal  voltage  hg  (the  former  obviously 
being  proportional  to  the  latter  in  the  shunt  machine).  The 
open-circuit  or  no-load  voltage  of  the  machine  is  then  kq. 

Drawing  gl  parallel  to  da  (assuming  constant  coefficient  of 
armature  reaction,  or  parallel  to  the  hypothenuse  of  the  triangle 
iq,  ir  at  voltage  og,  when  assuming  variable  armature  reaction), 
then  the  current  which  gives  voltage  gh  is  proportional  to  gl, 
that  is,  i  :  iQ  =  gl  :  da,  where  iQ  is  the  current  at  the  voltage  de. 

As  seen  from  Fig.  113,  a  maximum  value  of  current  exists 
which  is  less  if  the  brushes  are  shifted  than  at  constant  position 
of  brushes. 

From  the  load  characteristic  of  the  shunt  generator  the 
resistance  characteristic  is  plotted  in  Fig.  114;  that  is,  the  de- 
pendence of  the  terminal  voltage  upon  the  external  resistance 

„       terminal  voltage      ~  .      ^.  , 

R  =  — • — *-•     Curve  A   in   Fig.    114   corresponds   to 

current 


D.  C.  COMMUTATING  MACHINES 


211 


constant,  curve  B  to  varying  armature  reaction.  It  is  seen 
that  at  a  certain  definite  resistance  the  voltage  becomes  zero, 
and  for  lower  resistance  the  machine  cannot  generate  but  loses 
its  excitation. 

The  variation  of  the  terminal  voltage  of  the  shunt  generator 
with  the  speed  at  constant  field  resistance  is  shown  in  Fig.  115, 
at  no  load  as  A,  and  at  constant  current  i  as  B.  These  curves 
are  derived  from  the  preceding  ones.  They  show  that  below  a 
certain  speed,  which  is  much  higher  at  load  than  at  no  load,  the 


r 


50     100     150     200     250     300     350 

FIG.  113. — Shunt  generator  load  characteristic. 


machine  cannot  generate, 
and  cannot  be  realized. 


The  lower  part  of  curve  B  is  unstable 


Series  Generator 

72.  In  the  series  generator  the  field  excitation  is  proportional 
to  the  current  i,  and  the  saturation  curve  A  in  Fig.  116  can  thus 
be  plotted  with  the  current  i  as  abscissas.  Subtracting  ab  =  ir, 
the  resistance  drop,  from  the  voltage,  and  adding  bd  =  iq,  the 
armature  reaction,  gives  a  load  saturation  curve  or  external 
characteristic  B  of  the  series  generator.  The  terminal  voltage 
is  zero  at  no  load  or  open  circuit,  increases  with  the  load,  reaches 
a  maximum  value  at  a  certain  current,  and  then  decreases  again 
and  reaches  zero  at  a  certain  maximum  current,  the  current  of 
short  circuit. 

Curve  B  is  plotted  with  constant  coefficient  of  armature  reac- 
tion q.  Assuming  the  brushes  to  be  shifted  with  the  load  and 


212     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


0.1    0.2    0.3    0.4    0.5    0.6  0.8  1.0  1.2  1.1  1.6  1.8  2.0 

FIG.  114. — Shunt  generator  resistance  characteristic. 


/ 

1RO 

/ 

170 

/ 

/            A 

160 
150 
140 
130 

/ 

/ 

/ 

/ 

/ 

'  / 

/ 

/ 

/ 

/ 

no 

/ 

t 

/ 

100 

/ 

' 

90 
80 
70 
60 
50 
40 
30 

A 

/ 

4 

I 

/ 

/ 

/ 

v 

/ 

•*-  — 

— 

• 

A 

/ 

/ 

SO 

( 

10 

/ 

SPEE 

->• 

I 

0.2                  0.4                 0.6                  0.8                  1.0                  1.2                  1.4 

FIG.  115. — Shunt  generator  speed  characteristic  at  constant  field  circuit 

resistance. 


D.  C.  COMMUTATING  MACHINES 


213 


proportionally  to  the  load,  gives  curves  C,  D,  and  E,  which  are 
higher  at  light  load,  but  fall  off  faster  at  high  load.  A  still 
further  shift  of  brushes  near  the  maximum  current  value  even 
overturns  the  curve  as  shown  in  F.  Curves  E  and  F  correspond 
to  a  very  great  shift  of  brushes,  and  an  armature  demagnetizing 
effect  of  the  same  magnitude  as  the  field  excitation,  as  realized 
in  arc-light  machines,  in  which  the  last  part  of  the  curve  is  used 
to  secure  inherent  regulation  for  constant  current. 

The  resistance  characteristic,  that  is,  the  dependence  of  the 
current  and  of  the  terminal  voltage  of  the  series  generator  upon 


6000 


6000 


1   23   i   5   6   7   8   9   10  11  12  13   U  15  16  17  18  19 

FIG.  116. — Series  generator  saturation  curve  and  load  characteristic. 

the  external  resistance,  is  constructed  from  Fig.  116  and  plotted 
in  Fig.  117. 

BI  and  Bz  in  Fig.  117  are  terminal  volts  and  amperes  corre- 
sponding to  curve  B  in  Fig.  116,  #1,  Ez,  and  F%  volts  and  amperes 
corresponding  to  curves  E  and  F  in  Fig.  116. 

Above  a  certain  external  resistance  the  series  generator  loses 
its  excitation,  while  the  shunt  generator  loses  its  excitation 
below  a  certain  external  resistance. 


Compound  Generator 

73.  The  saturation  curve  or  magnetic  characteristic  A,  and 
the  load  saturation  curves  D  and  G  of  the  compound  generator, 
are  shown  in  Fig.  118  with  the  ampere-turns  of  the  shunt  field 


214     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


as  abscissas.     A  is  the  same  curve  as  in  Fig.  109,  while  D  and 
G  in  Fig.  118  are  the  corresponding  curves  of  Fig.  109  shifted  to 


7000 


50001 


8000 


1000 


\ 


RE 


ISTA.NCE 


ANCE 


\ 


200     400     600     800    1000    1200    1400    1600    180 

FIG.  117. — Series  generator  resistance  characteristic. 


;>OQOOMMS 


400 


3000    1000    5000    6000    7000    8000   9000 


FIG.  118. — Compound  generator  saturation  curve. 

the  left  by  the  distance  iqQ,  the  m.m.f.  of  ampere-turns  of  the 
series  field. 

At  constant  position  of  brushes  the  compound  generator,  when 


D.  C.  COMMUTATING  MACHINES 


215 


adjusted  for  the  same  voltage  at  no  load  and  at  full  load,  under- 
compounds  at  higher  and  over-compounds  at  lower  voltage,  and 
even  at  open  circuit  of  the  shunt  field  gives  still  a  voltage  op  as 
series  generator.  When  shifting  the  brushes  under  load,  at  lower 
voltage  a  second  point  g  is  reached  where  the  machine  compounds 
correctly,  and  below  this  point  the  machine  under-compounds 
and  loses  its  excitation  when  the  shunt  field  decreases  below  a 
certain  value;  that  is,  it  does  not  excite  itself  as  series  generator. 


B.  MOTORS 
Shunt  Motor 

74.  Three  speed  characteristics  of  the  shunt  motor  at  con- 
stant impressed  e.m.f.  e  are  shown  in  Fig.  116  as  A,  P,  Q,  corre- 
sponding to  the  points  d,  p,  q  of  the  motor  load  saturation  curve, 
Fig.  110.  Their  derivation  is  as  follows:  At  constant  impressed 


ir,0 
1100 


WOO 
£00 

(•00 

w 


50  100  150  200250  300  350  400  450  500 

FIG.  119. — Shunt  motor  speed  curves,  constant  impressed  e.m.f. 

e.m.f.  e  the  field  excitation  is  constant  and  equals  FQ,  and  at 
current  i  the  generated  e.m.f.  must  be  e  —  ir.  The  resultant 
field  excitation  is  F0  —  iqt  and  corresponding  hereto  at  constant 
speed  the  generated  e.m.f.  taken  from  saturation  curve  A  in 
Fig.  110  is  e\.  Since  it  must  be  e  —  ir,  the  speed  is  changed  in 

, .      e  -  ir 
the  proportion • 

At  a  certain  voltage  the  speed  is  very  nearly  constant,  the 
demagnetizing  effect  of  armature  reaction  counteracting  the 
effect  of  armature  resistance.  At  higher  voltage  the  speed  falls, 
at  lower  voltage  it  rises  with  increasing  current. 

In  Fig.  120  is  shown  the  speed  characteristic  of  the  shunt 


216     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

motor  as  function  of  the  impressed  voltage  at  constant  output, 
that  is,   constant  product,   current  times  generated  e.m.f.     If 

i  =  current    and   P  =  constant    output,    the    generated    e.m.f. 

p 
must  be  approximately  e\  =  — ,  and  thus  the  terminal  voltage 

e  =  e\  +  ir.     Proportional    hereto   is   the   field   excitation  FQ. 
The  resultant  m.m.f.  of  the  field  is  thus  F  =  FQ  —  iq,  and  corre- 
sponding thereto  from  curve  A  in  Fig.  Ill  is  derived  the  e.m.f.  eQ 
which  would  be  generated  at  constant  speed  by  the  m.m.f.  F. 
Since,  however,  the  generated  e.m.f.  must  be  e\}  the  speed  is 

changed  in  the  proportion  — • 

00 


REV. 
PER 

2000 
1800 
1600 
1400 
1200 
1000 
800 
COO 

[400 

200 

/ 

* 

/ 

/ 

/ 

s 

/ 

/ 

/ 

7 

nt 

/ 

/ 

1 

/ 

/ 

1 

/ 

^r 

S 

vor 

-s* 

20     10     60     80     100    120     110     160     ISO     200 

FIG.  120. — Shunt  motor  speed  curve,  variable  impressed  e.m.f. 

The  speed  rises  with  increasing  and  falls  with  decreasing  im- 
pressed e.m.f.  Still  further  decreasing  the  impressed  e.m.f., 
the  speed  reaches  a  minimum  and  then  increases  again,  but  the 
conditions  become  unstable. 


Series  Motor 

75.  The  speed  characteristic  of  the  series  motor  is  shown  in 
Fig.  121  at  constant  impressed  e.m.f.  e.  A  is  the  saturation 
curve  of  the  series  machine,  with  the  current  as  abscissas  and  at 
constant  speed.  At  current  i,  the  generated  e.m.f.  must  be 

/?    A  f 

e  —  ir,  and  the  speed  is  thus  —      '-  times  that,  for  which  curve  A 

Q\ 

is  plotted,  where  e\  =  e.m.f.  taken  from  saturation  curve  A. 


D.  C.  COMMUTATING  MACHINES 


217 


This  speed  curve  corresponds  to  a  constant  position  of  brushes 
midway  between  the  field  poles,  as  generally  used  in  railway 
motors  and  other  series  motors.  If  the  brushes  have  a  constant 
shift  or  are  shifted  proportionally  to  the  load,  instead  of  the 
saturation  curve  A  in  Fig.  121  a  curve  is  to  be  used  correspond- 
ing to  the  position  of  brushes,  that  is,  derived  by  adding  to  the 
abscissas  of  A  the  values  iq,  the  demagnetizing  effect  of  arma- 
ture reaction. 


10      60 

FIG.  121 


100     120    _110    160     ISO 

-Series  motor  speed  curve. 


The  torque  of  the  series  motor  is  shown  also  in  Fig.  121, 
derived  as  proportional  to  A  X  i,  that  is,  current  X  magnetic 
flux. 

Compound  Motors 

76.  Compound  motors  can  be  built  with  cumulative  com- 
pounding and  with  differential  compounding. 

Cumulative  compounding  is  used  to  a  considerable  extent,  as 
in  elevator  motors,  etc.,  to  secure  economy  of  current  in  starting 
and  at  high  loads  at  the  sacrifice  of  speed  regulation;  that  is,  a 
compound  motor  with  cumulative  series  field  stands  in  its  speed 
and  torque  characteristic  intermediate  between  the  shunt  motor 
and  the  series  motor. 


218     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Differential  compounding  is  used  to  secure  constancy  of  speed 
with  varying  load,  but  to  a  small  extent  only,  since  the  speed 
regulation  of  a  shunt  motor  can  be  made  sufficiently  close,  as  was 
shown  in  the  preceding. 

Conclusion 

77.  The  preceding  discussion  of  commutating  machine  types 
can  obviously  be  only  very  general,  showing  the  main  character- 
istics of  the  curves,  while  the  individual  curves  can  be  modified 
to  a  considerable  extent  by  suitable  design  of  the  different  parts 
of  the  machine  when  required  to  derive  certain  results,  as,  for 
instance,  to  extend  the  constant-current  part  of  the  series  gen- 
erator; or  to  derive  a  wide  range  of  voltage  at  stability,  that  is, 
beyond  the  bend  of  the  saturation  curve  in  the  shunt  generator; 
or  to  utilize  the  range  of  the  shunt  generator  load  characteristic 
at  the  maximum  current  point  for  constant-current  regulation; 
or  to  secure  constancy  of  speed  in  a  shunt  motor  at  varying 
impressed  e.m.f.,  etc. 

The  use  of  the  commutating  machine  as  direct-current  con- 
verter has  been  omitted  from  the  preceding  discussion.  By 
means  of^  one  or  more  alternating-current  compensators  or 
autotransformers,  connected  to  the  armature  by  collector 
rings,  the  commutating  machine  can  be  used  to  double  or  halve 
the  voltage,  or  convert  from  one  side  of  a  three-wire  system  to 
the  other  side  and,  in  general,  to  supply  a  three-wire  Edison 
system  from  a  single  generator.  Since,  however,  the  direct- 
current  converter  and  three-wire  generator  exhibit  many  fea- 
tures similar  to  those  of  the  synchronous  converter,  as  regards 
the  absence  of  armature  reaction,  the  reduced  armature  heat- 
ing, etc.,  they  will  be  discussed  as  an  appendix  to  the  synchro- 
nous converter. 

XV.  APPENDIX 
ALTERNATING-CURRENT  COMMUTATOR   MOTOR 

78.  Since  in  the  series  motor  and  in  the  shunt  motor  the 
direction  of  the  rotation  remains  the  same  at  a  reversal  of  the 
impressed  voltage,  these  motors  can  be  operated  by  an  alternat- 
ing voltage,   as    alternating-current    motors,   by   making   such 
changes  in  the  materials,  proportioning  and  design,  as  the  al- 
ternating nature  of  the  current  requires. 


D.  C.  COMMUTATING  MACHINES  219 

In  the  alternating-current  commutator  motor,  the  field  struc- 
ture as  well  as  the  armature  must  be  laminated,  since  the  mag- 
netic flux  is  alternating. 

The  alternation  of  the  field  flux  induces  an  e.m.f.  of  self 
induction  in  the  field  winding.  In  the  shunt  motor,  this  causes 
the  field  exciting  current  and  with  it  the  magnetic  field  flux  to 
lag  and  thereby  to  be  out  of  phase  with  the  armature  current 
which,  to  represent  work,  must  essentially  be  an  energy  current, 
and  thereby  reduces  output  and  efficiency  and  hence  requires 
some  method  of  compensation,  as  capacity  in  series  with  the 
field  winding  or  excitation  of  the  field  from  a  quadrature  phase 
of  voltage.  In  the  series  motor  the  self-inductance  of  the  field 
causes  the  main  current  to  lag  behind  the  impressed  voltage  and 
thereby  lowers  the  power-factor  of  the  motor.  Thus,  to  get 
good  power-factor,  the  field  self-inductance  must  be  made  low, 
that  is,  the  field  as  weak  and  the  armature  as  strong  as  possible. 
With  such  a  strong  armature,  and  weak  field,  the  commutating 
pole  is  not  sufficient  to  control  magnetic  distortion  by  the  arma- 
ture reaction,  and  complete  compensation  by  a  distributed 
compensating  winding,  as  Fig.  102,  page  190,  is  required. 

79.  When  in  the  position  of  commutation  the  armature 
coil  is  short-circuited  by  the  commutator  brush,  it  encloses  the 
full  field  flux  and  thus  for  a  moment  no  e.m.f.  is  induced  in  the 
armature  coil  by  its  rotation  through  the  field  flux,  and  in  the 
continuous  current  machine  the  coil  is  without  voltage  except 
whatever  voltage  may  be  intentionally  produced  by  the  com- 
mutating flux.  In  the  alternating-current  motor,  however,  the 
field  flux  induces  voltage  also  in  the  armature  coil  by  its 
alternation,  and  this  voltage  is  a  maximum  in  the  position  of 
commutation,  and  when  short-circuited  by  the  commutator 
brush  tends  to  produce  an  excessive  current  and  cause  spark- 
ing. No  position  exists  on  the  commutator  of  the  alternating- 
current  motor  where  the  armature  coil  does  not  contain  an 
induced  e.m.f.,  but  in  the  position  midway  between  the  brushes 
the  e.m.f.  induced  by  the  rotation  through  the  magnetic 
field  is  a  maximum;  in  the  position  of  commutation  the  e.m.f. 
induced  by  the  alternation  of  the  field  flux  is  a  maximum.  To 
overcome  the  destructive  sparking  caused  by  the  short  circuit 
of  the  latter  e.m.f.  by  the  commutator  brush  is  the  problem  of 
making  a  successful  alternating-current  commutator: 

1.  Inducing  an  opposite  e.m.f.  by  a  commutating  field.     As 


220     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

the  e.m.f.  induced  by  the  alternation  of  the  main  field  is  in 
quadrature  with  the  main  field,  and  the  e.m.f.  induced  by  the 
rotation  through  the  commutating  field  is  in  phase  with  it,  the 
commutating  field  must  be  in  quadrature  with  the  main  field. 
By  properly  proportioning  this  commutating  field,  as  in  the 
series  repulsion  motor,  completely  sparkless  commutation  can 
be  produced  at  speed.  However,  at  standstill  and  low  speeds 
this  method  fails,  as  the  voltage  induced  by  the  rotation  through 
the  commutating  field  becomes  zero  at  standstill. 

2.  Reducing  the  short-circuit  current  by  high  resistance  leads 
between  commutator  and  armature  coil.     This  only  mitigates 
the  trouble,  but  due  to  the  voltage  drop  in  the  lead  resistance 
tends  to  increase  sparking  at  speed.     Also,  the  excessive  con- 
centration of  heat  in  the  commutating  leads  in  the  moment  of 
starting  tends  to  destroy  them  if  the  motor  does  not  quickly 
start. 

3.  Narrow  brushes,  to  reduce  the  duration  of  short  circuit. 

4.  Low  impressed   frequency,  so  as  to   give   low  values  to 
the  induced  e.m.f.     This  is  the  cause  of  the  desire  for  abnormally 
low  frequencies,  as  15  and  even  8  cycles,  in  alternating-current 
railway  electrification. 

5.  Low  magnetic    flux  per  pole.     This    is  the    reason  why 
alternating-current  commutator  motors  of  large  power  usually 
have  such  a  large  number  of  poles. 

These  very  severe  limitations  of  the  design  of  alternating-cur- 
rent commutating  motors  are  the  reason  why  such  motors  have 
found  only  limited  application,  except  in  smaller  sizes. 

80.  Alternating-current  motors  are  usually  single-phase,  since 
the  possibility  of  commutation  control  makes  the  single-phase 
easier  than  a  polyphase  design.     In  the  single-phase  motor,  the 
magnetic  field  flux  is  constant  in  direction,  and  the  direction 
in  quadrature  to  the  main  field  flux  thus  is  available  for  pro- 
ducing a  suitable  commutating  flux.     In  the  polyphase  motor, 
however,  the  magnetic  flux  rotates,  assuming  successively  all 
directions,  and  thus  no  commutating  flux  can  be  used.     For  this 
reason,   designs   of  polyphase   commutator   motors  have  been 
made  in  which  the  different  (2  and  3)  phases  are  kept  separate, 
and  spaces  left  between  them  for  accommodating  commutating 
fluxes. 

81.  Alternating-current  commutator  motors  are  used: 

1.  In  railroading,  for  securing  the  advantage  of  the  higher 


D.  C.  COMMUTATING  MACHINES  221 

economy  of  high  voltage  alternating-current  transmission  and 
distribution.  For  railroading  generally  the  series  motor  type  is 
used,  either  the  plain  compensated  series  motor,  or  inductive 
modifications  thereof,  as  the  repulsion  motor  etc.  In  the  repul- 
sion motor  the  armature,  instead  of  being  connected  in  series 
with  field  and  compensating  winding,  is  closed  on  itself  and  thus 
traversed  by  a  secondary  current  induced  by  the  compensating 
winding  as  primary  that  is,  the  armature  is  connected  inductively 
in  series. 

2.  As    constant-speed    motor    where     considerable    starting 
torque  is  required,  as  for  elevators,  hoists,  etc.,  and  in  general 
as  self-starting  single-phase  motors.     For  this  purpose,  com- 
binations of  repulsion  and  induction  type  or  of  series  and  in- 
duction type  are  used. 

3.  As  adjustable  speed,  alternating-current  motor  of  single- 
phase  and  of  polyphase  type.     The  synchronous  motor  and 
the  induction  motor  both  are  constant  and  fixed  speed,  the 
former  synchronous,  the  latter  near  synchronous.     Operating 
the  induction  motor  materially  below  synchronism,  by  arma- 
ture resistance,  is  inefficient  and  gives  a  speed  which  varies  with 
the  load.     By  changing  the  number  of  poles,  or  by  concatena- 
tion,   multi-speed  induction    motors    can    be    produced.     The 
gradual  speed  adjustment,  as  given  by  field  control  of  direct- 
current  motors,   requires,  however,  a  commutator  on  the  al- 
ternating-current motor.     If  into  the  secondary  of  the  induction 
motor  an  e.m.f.  is  introduced,  the  speed  of  the  motor  can  be 
varied  by  varying  the  introduced  e.m.f.;  and  lowered,  if  this 
e.m.f.  is  in  opposition;  raised  beyond  synchronism,  if  this  e.m.f. 
is  in  the  same  direction  as  the  e.m.f.  induced  in  the  motor 
secondary.     As,  however,  the   e.m.f.  induced  in  the  induction 
motor  secondary  is  of  the  frequency  of  slip,  the  speed  controlling 
e.m.f.  must  either  be  supplied  through  the  commutator  or  de- 
rived from  a  low  frequency  commutating  machine  as  source. 

4.  For  power-factor  compensation.     In  an  inductive  circuit, 
the  current  lags  behind  the  voltage  or,  what  is  the  same,  the 
voltage  leads  the  current,  and   the  power-factor  thus  can  be 
raised  by  compensation  either  by  introducing  a  leading  current, 
as  from  condenser  or  overexcited  synchronous  motor,  or  by  in- 
troducing a  lagging  voltage.     In  the  commutating  machines, 
the  voltage  induced  in  the  armature  by  its  rotation  is  in  phase 
with  the  field  magnetism,  and  by  lagging  the  field  exciting  current, 


222     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

the  commutating  machines  thus  can  be  made  to  give  a  lagging 
voltage,  that  is,  to  compensate  for  low  power-factor  due  to  lagging 
current.  Thus,  by  inserting  such  a  commutating  machine  into 
the  secondary  of  an  induction  machine,  the  latter  can  be  made 
to  give  unity  power-factor  or  even  leading  current. 

Such  phase  compensation  is  frequently  used  in  alternating- 
current  commutator  motors  to  get  good  power-factor.  Thus  in 
the  series  motor,  by  shunting  the  field  by  a  non-inductive  re- 
sistance, and  thereby  lagging  the  field  exciting  component  of 
the  current  and  with  it  the  field  flux  and  the  voltage  induced 
in  the  armature  by  its  rotation,  behind  the  main  current,  the 
series  motor  can  at  higher  speeds  be  made  to  give  unity  power- 
factor.  At  low  speeds,  such  complete  compensation  is  not 
possible,  as  the  compensating  voltage  is  proportional  to  the 
speed. 


C.  SYNCHRONOUS  CONVERTERS 
I.  General 

82.  For  long-distance  transmission,  and  to  a  certain  extent 
also  for  distribution,  alternating  currents,  either  polyphase  or 
single-phase,  are  extensively  used.  For  many  applications, 
however,  as  especially  for  electrolytic  work,  direct  currents  are 
required,  and  are  usually  preferred  also  for  electrical  railroading 
and  for  low-tension  distribution  on  the  Edison  three- wire  system. 
Thus,  where  power  is  derived  from  an  alternating  system, 
transforming  devices  are  required  to  convert  from  alternating 
to  direct  current.  This  can  be  done  either  by  a  direct-current 
generator  driven  by  an  alternating  synchronous  or  induction 
motor,  or  by  a  single  machine  consuming  alternating  and  pro- 
ducing direct  current  in  one  and  the  same  armature.  Such  a 
machine  is  called  a  converter,  and  combines,  to  a  certain  extent, 
the  features  of  a  direct-current  generator  and  an  alternating 
synchronous  motor,  differing,  however,  from  either  in  other 
features. 

Since  in  the  converter  the  alternating  and  the  direct  current 
are  in  the  same  armature  conductors,  their  e.m.fs.  stand  in  a 
definite  relation  to  each  other,  which  is  such  that  in  practically 
all  cases  step-down  transformers  are  necessary  to  generate  the 
required  alternating  voltage. 

Comparing  thus  the  converter  with  the  combination  of  syn- 
chronous or  induction  motor  and  direct-current  generator,  the 
converter  requires  step-down  transformers;  the  synchronous 
motor,  if  the  alternating  line  voltage  is  considerably  above 
10,000  volts,  generally  requires  step-down  transformers  also; 
with  voltages  of  1000  to  10,000  volts,  however, ,  usually  the 
synchronous  motor  and  frequently  the  induction  motor  can  be 
wound  directly  for  the  line  voltage  and  stationary  transformers 
saved.  Thus  on  the  one  side  we  have  two  machines  with  or 
sometimes  without  stationarytransformers,  on  the  other  side 
a  single  machine  with  transformers. 

Regarding  the  reliability  of  operation  and  first  cost,  obviously 
a  single  machine  is  preferable. 

223 


224     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Regarding  efficiency,  it  is  sufficient  to  compare  the  converter 
with  the  synchronous-motor-direct-current-generator  set,  since 
the  induction  motor  is  usually  less  efficient  than  the  syn- 
chronous motor.  The  efficiency  of  stationary  transformers  of 
large  size  varies  from  97  per  cent,  to  98  per  cent.,  with  an  average 
of  97.5  per  cent.  That  of  converters  or  of  synchronous  motors 
varies  between  91  per  cent,  and  95  per  cent.,  with  93  per  cent,  as 
average,  and  that  of  the  direct-current  generator  between  90  per 
cent,  and  94  per  cent.,  with  92  per  cent,  as  average.  Thus  the 
converter  with  its  step-down  transformers  will  give  an  average 
efficiency  of  90.7  per  cent.,  a  direct-current  generator  driven  by 
synchronous  motor  with  step-down  transformers  an  efficiency 
of  83.4  per  cent.,  without  step-down  transformers  an  efficiency  of 
85.6  per  cent.  Hence  the  converter  is  more  efficient,  and  there- 
fore is  almost  always  preferred. 

Mechanically  the  converter  has  the  advantage  that  no  transfer 
of  mechanical  energy  takes  place,  since  the  torque  consumed  by 
the  generation  of  the  direct  current  and  the  torque  produced  by 
the  alternating  current  are  applied  at  the  same  armature  con- 
ductors, while  in  a  direct-current  generator  driven  by  a  syn- 
chronous motor  the  power  has  to  be  transmitted  mechanically 
through  the  shaft. 

EC.  Ratio  of  e.m.fs.  and  of  Currents 

83.  In  its  structure  the  synchronous  converter  consists  of 
a  closed-circuit  armature,  revolving  in  a  direct-current  excited 
field,  and  connected  to  a  segmental  commutator  as  well  as  to 
collector  rings.  Structurally  it  thus  differs  from  a  direct- 
current  machine  by  the  addition  of  the  collector  rings,  from 
certain  (now  very  little  used)  forms  of  synchronous  machines  by 
the  addition  of  the  segmental  commutator. 

In  consequence  hereof,  regarding  types  of  armature  windings 
and  of  field  windings,  etc.,  the  same  rule  applies  to  the  converter 
as  to  all  commutating  machines,  except  that  in  the  converter  the 
total  number  of  armature  coils  with  a  series-wound  armature, 
and  the  number  of  armature  coils  per  pair  of  poles  with  a  multiple- 
wound  armature,  must  be  divisible  by  the  number  of  phases, 
and  that  multiple  spiral  and  reentrant  windings  are  difficult  to 
apply. 

Regarding  the  wave  shape  of  the  alternating  counter-gener- 


SYNCHRONOUS  CONVERTERS        225 

ated  e.m.f.,  similar  considerations  apply  as  for  a  synchronous 
machine  with  closed-circuit  armature;  that  is,  the  generated 
e.m.f.  usually  approximates  a  sine  wave,  due  to  the  multi-tooth 
distributed  winding. 

Thus,  in  the  following,  only  those  features  will  be  discussed 
in  which  the  synchronous  converter  differs  from  the  commu- 
tating  machines  and  synchronous  machines  treated  in  the 
preceding  chapters. 

Fig.  122  represents  diagrammatically  the  commutator  of  a 
direct-current  machine  with  the  armature  coils  A  connected  to 
adjacent  commutator  bars.  The  brushes  are  BiB2,  and  the  field 
poles  FiF2. 

If  now  two  oppositely  located  points  a ia2  of  the  commutator 
are  connected  with  two  collector  rings  DiD2,  it  is  obvious  that 


'8 

FIG.  122. — Single-phase  converter  commutator. 

the  e.m.f.  between  these  points  aia2,  and  thus  between  the 
collector  rings  DiZ>2,  will  be  a  maximum  in  the  moment  when 
the  points  aia2  coincide  with  the  brushes  BiB2,  and  is  in  this 
moment  equal  to  the  direct  voltage  E  of  this  machine.  While 
the  points  ai<z2  move  away  from  this  position,  the  difference  of 
potential  between  aj,  and  a2  decreases  and  becomes  zero  in  the 
moment  where  aia2  coincide  with  the  direction  of  the  field  poles 
FiF2.  In  this  moment  the  difference  in  potential  between  di 
and  a2  reverses  and  then  increases  again,  reaching  equality  with 
E,  but  in  opposite  direction,  when  ai  and  a2  coincide  with  the 
brushes  B2  and  BI;  that  is,  between  the  collector  rings  DI  and 
D2  an  alternating  voltage  is  produced  whose  maximum  value 
equals  the  direct-current  electromotive  force  E,  and  which  makes 
a  complete  period  for  every  revolution  of  the  machine  (in  a 


226     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

bipolar  converter,  or  p  periods  per  revolution  in  a  machine  of 
2  p  poles). 

Hence,  this  alternating  e.m.f.  is 

e  =  E  sin  2  trft, 

where  /  =  frequency  of  rotation,  E  =  e.m.f.  between  brushes 
of  the  machine;  thus,  the  effective  value  of  the  alternating 
e.m.f.  is 

F  ;'* 
El 


84.  That    is,    a    direct-current    machine    produces    between 
two  collector  rings  connected  with  two  opposite  points  of  the 

commutator  an  alternating  e.m.f.  of  —-=  X  the  direct-current 

v2 

voltage,  at  a  frequency  equal  to  the  fre- 
quency of  rotation.  Since  every  alternating- 
current  generator  is  reversible,  such  a  direct- 
current  machine  with  two  collector  rings, 
when  supplied  with  an  alternating  e.m.f.  of 
1 

X  the  direct-current  voltage  at  the  fre- 


quency of  rotation,  will  run  as  synchronous 
motor,  or  if  at  the  same  time  generating 
direct  current,  as  synchronous  converter. 

convert™mSataoSr.  ^ce,  neglecting  losses  and  phase  dis- 
placement,  the  output  of  the  direct-current 

side  must  be  equal  to  the  input  of  the  alternating-current  side, 

1 

and  the  alternating  voltage  in  the  single-phase  converter  is     ,~ 

X  E,  the  alternating  current  must  be  =  \/2  X  /,  where  /  = 
direct-current  output. 

If  now  the  commutator  is  connected  to  a  further  pair  of  col- 
lector rings,  D3D4  (Fig.  123),  at  the  points  a3  and  a4  midway  be- 
tween ai  and  a2,  it  is  obvious  that  between  Z>3  and  D4  an  alter- 
nating voltage  of  the  same  frequency  and  intensity  will  be 
produced  as  between  DI  and  D2,  but  in  quadrature  therewith, 
since  at  the  moment  where  a3  and  a4  coincide  with  the  brushes 
BiB2  and  thus  receive  the  maximum  difference  of  potential,  ai 
and  az  are  at  zero  points  of  potential. 

Thus  connecting  four  equidistant  points  a\,  a2,  0,3,  a4  of  the 


SYNCHRONOUS  CONVERTERS 


227 


direct-current  generator  to  four  collector  rings  D\,  D2,  D3,  D4, 
gives  a  four-phase  converter  of  the  e.m.f. 

EI  =  —=  E  per  phase. 

v  2 

The  current  per  phase  is  (neglecting  losses  and  phase  displace- 
ment) 


since  the  alternating  power,  2  EJi,  must  equal  the  direct-current 
power,  EI. 

Connecting  three  equidistant  points  of  the  commutator  to 
three  collector  rings  as  in  Fig.  124  gives  a  three-phase  converter. 

85.  In  Fig.  125  the  three  e.m.fs.  between  the  three  collector 
rings  and  the  neutral  point  of  the  three-phase  system  (or  Y 
voltages)  are  represented  by  the  vectors  OEi,  OEZ,  OEs,  thus 


FIQ.  124.— Three-phase  syn- 
chronous converter. 


FIQ.  125. — E.m.f.  diagram  of 
three-phase  converter. 


the  e.m.f.  between  the  collector  rings  or  the  delta  voltages  by 
vectors  EiE2,  E2E3,  and  E$E\.  The  e.m.f.  OEi  is,  however, 
nothing  but  half  the  e.m.f.  EI  in  Fig.  122,  of  the  single-phase 

Tjl 

converter,   that  is,  =  -  =  -     Hence  the  Y  voltage,  or  voltage 
2  v  2 

between  collector  ring  and  neutral  point  or  center  of  the  three- 
phase  voltage  triangle,  is 


—  -=  =  0.354  E. 

2V2 


and  thus  the  delta  voltage  is 
E'  =  El  V3 


0.612  E. 


228     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Since  the  total  three-phase   power  3  IiEi  equals  the  total 
continuous-current  power  IE,  it  is 


In  general,  in  an  n-phase  converter,  or  converter  in  which 
n  equidistant  points  of  the  commutator  (in  a  bipolar  machine, 
or  n  equidistant  points  per  pair  of  poles  in  a  multipolar  machine 
with  multiple-wound  armature)  are  connected  to  n  collector 
rings,  the  voltage  between  any  collector  ring  and  the  common 
neutral,  or  star  voltage,  is 


consequently  the  voltage  between  two  adjacent  collector  rings, 
or  ring  voltage,  is 


s' 


E  sin- 


V2 


since  —  is  the  angular  displacement  between  two  adjacent  col- 

lector rings.     Herefrom  the  current  per  line,  or  star  current,  is 
found  as 

2V27 


and  the  current  from  line  to  line,  or  from  collector  ring  to  ad- 
jacent collector  ring,  or  ring  current,  is 

V2/ 


r  = 


.       7T 

n  sin  — 
n 


86.  As  seen  in  the  preceding,  in  the  single-phase  converter 
consisting  of  a  closed-circuit  armature  tapped  at  two  equi- 
distant points  to  the  two  collector  rings,  the  alternating  voltage 

is  —-=  times  the  direct-current  voltage,  and  the  alternating  cur- 
V  2  _ 

rent  \/2  times  the  direct  current.  While  such  an  arrangement  of 
the  single-phase  converter  is  the  simplest,  requiring  only  two 
collector  rings,  it  is  undesirable,  especially  for  larger  machines, 
on  account  of  the  great  total  and  especially  local  7V  heating  in 
the  armature  conductors,  as  will  be  shown  in  the  following,  and 


SYNCHRONOUS  CONVERTERS        229 

due  to  the  waste  of  e.m.f.,  since  in  the  circuit  from  collector 
ring  to  collector  ring  the  e.m.fs.  generated  in  the  coils  next  to 
the  leads  are  wholly  or  almost  wholly  opposite  to  each  other. 

The  arrangement  which  I  have  called  the  two-circuit  single- 
phase  converter,  and  which  is  diagrammatically  shown  in  Fig. 
126,  is  therefore  preferable.  The  step-down  transformer  T 
contains  two  independent  secondary  coils  A  and  B,  of  which 
one,  Ay  feeds  into  the  armature  over  conductor  rings  DiD2  and 
leads  dia2,  the  other,  B,  over  collector  rings  D3Z>4  and  leads  a3a4, 
so  that  the  two  circuits  aiaz  and  a3a4  are  in  phase  with  each 
other,  and  each  spreads  over  120  deg.  arc  instead  of  180  deg. 
arc  as  in  the  single-circuit  single-phase  converter. 


a* 
FIG.  126.  —  Two-circuit  single-phase  converter. 

In  consequence  thereof,  in  the  two-circuit  single-phase  con- 
verter the  alternating  counter-generated  e.m.f.  bears  to  the  con- 
tinuous-current e.m.f.  the  same  relation  as  in  the  three-phase 
converter,  that  is, 


and  from  the  equality  of  alternating-  and  direct-current  power, 

2  /i#i  =  IE, 
it  follows  that  each  of  the  two  single-phase  supply  currents  is 

-v/2 
If  =  -I  =  0.8177. 


It  is  seen  that  in  this  arrangement  one-third  of  the  armature, 
from  ai  to  a3  and  from  a2  to  a4,  carries  the  direct  current  only, 
the  other  two-thirds,  from  ai  to  a2  andfroma3to  a4,  the  differential 


current. 

x5 


230     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

A  six-phase  converter  is  usually  fed  from  a  three-phase  system 
by  three  transformers  or  one  three-phase  transformer.  These 
transformers  can  either  have  each  one  secondary  coil  only  of 

E 
twice  the  star  or  7  voltage,  =  —T=>  which  connects  with  its  two 

terminals  two  collector  rings  leading  to  two  opposite  points  of 
the  armature,  or  each  of  the  step-down  transformers  contains 
two  independent  secondary  coils,  and  each  of  the  two  sets  of 
secondary  coils  is  connected  in  three-phase  delta  or  F,  but  the 
one  set  of  coils  reversed  with  regard  to  each  other,  thus  giving 
two  three-phase  systems  which  join  to  a  six-phase  system. 

The  different  transformer  connections  then  are  distinguished 
as  "diametrical/7  "double  delta"  and  "double  F." 

For  further  arrangements  of  six-phase  transformation,  see 
"Theory  and  Calculation  of  Alternating-current  Phenomena/7 
fourth  edition,  Chapter  XXXVI. 

The  table  below  gives,  with  the  direct-current  voltage  and 
direct  current  as  unit,  the  alternating  voltages  and  currents  of 
the  different  converters. 


-u  •** 
11 

11 

1! 

V 

§ 
43 
A 

i 

1 
f 

1 
A 

dve-phase 

1 

"  w 

fl.S 

§2 

,rj 

§ 

i 

•a 

08  °° 

hj 

H 

£ 

00 

H    * 

i 

1 

i 

1 

i 

1 

1 

i 

Volts  between  collector 

2\/2 

2V2 

2V2 

2-S/2 

2V2 

2V2 

2\/2 

ring  and  neutral  point. 

=  0  .  354 

=  0.354 

=  0  .  354 

=  0.354 

=  0.354 

=  0  .  354 

=  0  .  354 

1 

V3 

V3 

1 

.       Tf 

sin  — 

Volts  between  adjacent 
collector  rings  

1.0 

V2 
=  0.707 

2V2 
=  0.612 

2V2 
=  0.612 

^=0.5 

2V2 
=  0.354 

0.183 

n 

vl 

Amperes  per  line  

V2 

2-S/2 

1 

V2 

2"V^2 

V2 

V3 

3 

V2 

3 

n 

1.0 

=  1.414 

=  0.817 

=  0.943 

=  0  .  707 

=  0.472 

0.236 

Amperes  between  ad- 
jacent lines  

V2 

V2 
V3 

2V  2 
3V3 

V2 
3 

.     n 

.       IT 

=  1.414 

=  0.817 

=  0.545 

^  =  0.5 

=  0.472 

0.455 

n 

These  currents  give  only  the  power  component  of  alternating 
current  corresponding  to  the  direct-current  output.  Added 
thereto  is  the  current  required  to  supply  the  losses  in  the  machine, 
that  is,  to  rotate  it,  and  the  wattless  component  if  a  phase  dis- 
placement is  produced  in  the  converter. 


SYNCHRONOUS  CONVERTERS        231 

HI.  Variation  of  the  Ratio  of  Electromotive  Forces 

87.  The  preceding  ratios  of  e.m.fs.  apply  strictly  only  to  the 
generated  e.m.fs.  and  that  under  the  assumption  of  a  sine  wave 
of  alternating  generated  e.m.f. 

The  latter  is  usually  a  sufficiently  close  approximation,  since 
the  armature  of  the  converter  is  a  multi-tooth  structure,  that  is, 
contains  a  distributed  winding. 

The  ratio  between  the  difference  of  potential  at  the  commu- 
tator brushes  and  that  at  the  collector  rings  of  the  converter 
usually  differs  somewhat  from  the  theoretical  ratio,  due  to  the 
e.m.f.  consumed  in  the  converter  armature,  and  in  machines 
converting  from  alternating  to  continuous  current,  also  due  to 
the  shape  of  the  impressed  wave. 

When  converting  from  alternating  to  direct  current,  under 
load  the  difference  of  potential  at  the  commutator  brushes  is 
less  than  the  generated  direct  e.m.f.,  and  the  counter-generated 
alternating  e.m.f.  less  than  the  impressed,  due  to  the  voltage 
consumed  by  the  armature  resistance. 

If  the  current  in  the  converter  is  in  phase  with  the  impressed 
e.m.f.,  armature  self-inductance  has  little  effect,  but  reduces  the 
counter-generated  alternating  e.m.f.  below  the  impressed  with 
a  lagging  and  raises  it  with  a  leading  current,  in  the  same  way  as 
in  a  synchronous  motor. 

Thus  in  general  the  ratio  of  voltages  varies  somewhat  with  the 
load  and  with  the  phase  -relation,  and  with  constant  impressed 
alternating  e.m.f.  the  difference  of  potential  at  the  commutator 
brushes  decreases  with  increasing  load,  decreases  with  decreasing 
excitation  (lag),  and  increases  with  increasing  excitation  (lead). 

When  converting  from  direct  to  alternating  current  the  reverse 
is  the  case. 

The  direct-current  voltage  stands  in  definite  proportion  only 
to  the  maximum  value  of  the  alternating  voltage  (being  equal 
to  twice  the  maximum  star  voltage),  but  to  the  effective  value 
(or  value  read  by  voltmeter)  only  in  so  far  as  the  latter  depends 

upon  the  former,  being  =  — - 7=  maximum  value  with  a  sine  wave. 

Thus  with  an  impressed  wave  of  e.m.f.  giving  a  different  ratio 
of  maximum  to  effective  value,  the  ratio  between  direct  and 
alternating  voltage  is  changed  in  the  same  proportion  as  the  ratio 
of  maximum  to  effective;  thus,  for  instance,  with  a  flat-topped 


232     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

wave  of  impressed  e.m.f.,  the  maximum  value  of  alternating 
impressed  e.m.f.,  and  thus  the  direct  voltage  depending  there- 
upon, are  lower  than  with  a  sine  wave  of  the  same  effective 
value,  while  with  a  peaked  wave  of  impressed  e.m.f.  they  are 
higher,  by  as  much  as  10  per  cent,  in  extreme  cases. 

In  determining  the  wave  shape  of  impressed  e.m.f.  at  the  con- 
verter terminals,  not  only  the  wave  of  generator  e.m.f.,  but  also 
that  of  the  converter  counter  e.m.f.,  may  be  instrumental.  Thus, 
with  a  converter  connected  directly  to  a  generating  system  of  very 
large  capacity,  the  impressed  e.m.f.  wave  will  be  practically 
identical  with  the  generator  wave,  while  at  the  terminals  of  a 
converter  connected  to  the  generator  over  long  lines  with  re- 
active coils  or  inductive  regulators  interposed,  the  wave  of  im- 
pressed e.m.f.  may  be  so  far  modified  by  that  of  the  counter  e.m.f. 
of  the  converter  as  to  resemble  the  latter  much  more  than  the 
generator  wave,  and  thereby  the  ratio  of  conversion  may  be  quite 
different  from  that  corresponding  to  the  generator  wave. 

Furthermore,  for  instance,  in  three-phase  converters  fed  by 
ring  or  delta  connected  transformers,  the  star  e.m.f.  at  the  con- 
verter terminals,  which  determines  the  direct  voltage,  may 
differ  from  the  star  e.m.f.  impressed  by  the  generator,  by  con- 
taining different  third  and  ninth  harmonics,  which  cancel  when 
compounding  the  star  voltages  to  the  delta  voltage,  and  give 
identical  delta  voltages,  as  required. 

Hence,  the  ratios  of  e.m.fs.  given  in  Section  II  have  to  be 
corrected  by  the  drop  of  voltage  in  the  armature,  and  have  to 
be  multiplied  by  a  factor  which  is  \/2  times  the  ratio  of  effective 
to  maximum  value  of  impressed  wave  of  star  e.m.f.  (\/2  being 
the  ratio  of  maximum  to  effective  of  the  sine  wave  on  which  the 
ratios  in  Section  II  were  based),  that  is,  by  a  "form  factor" 
of  the  e.m.f.  wave. 

With  an  impressed  wave  differing  from  the  sine  shape,  there  is  a 
current  of  higher  frequency,  but  generally  of  negligible  mag- 
nitude, through  the  converter  armature,  due  to  the  difference 
between  impressed  and  counter  e.m.f.  wave. 

IV.  Armature  Current  and  Heating 

88.  The  current  in  the  armature  conductors  of  a  converter 
is  the  difference  between  the  alternating-current  input  and  the 
direct-current  output. 


SYNCHRONOUS  CONVERTERS 


233 


In  Fig.  127,  ai,  a2  are  two  adjacent  leads  connected  with  the 
collector  rings  DI,  D2  in  an  n-phase  converter.  The  alternating 
e.m.f.  between  a\  and  a2,  and  thus  the  power  component  of  the 
alternating  current  in  the  armature  section  between  a\  and  a2, 
will  reach  a  maximum  when  this  section  is  midway  between  the 
brushes  BI  and  Bz,  as  shown  in 
Fig.  127. 

The  direct  current  in  every 
armature  coil  reverses  at  the  mo- 
ment when  the  coil  passes  under 
brush  BI  or  B2,  and  is  thus  a  rec- 
tangular alternating  current  as 
shown  in  Fig.  128  as  7.  At  the 
moment  when  the  power  com- 
ponent of  the  alternating  current 
is  a  maximum,  an  armature  coil 
d  midway  between  two  adjacent 
alternating  leads  ai  and  a2  is  midway  between  the  brushes 
BI  and  B2}  as  in  Fig.  127,  and  is  thus  in  the  middle  of  its  rectan- 
gular continuous-current  wave,  and  consequently  in  this  coil 
the  power  component  of  the  alternating  current  and  the  rectan- 
gular direct  current  are  in  phase  with  each  other,  but  opposite,  as 


FIG.  127. — Diagram  for  study  of 
armature  heating  in  synchronous 
converters. 


FIG.  128. — Direct  current  and  alternating  current  in  armature  coil  d, 

Fig.  127. 


FIG.  129. — Resultant  current  in  coil  d,  Fig.  127. 

shown  in  Fig.  128  as  7i  and  /,  and  the  actual  current  is  their 
difference,  as  shown  in  Fig.  129. 

In  successive  armature  coils  the  direct  current  reverses  suc- 
cessively; that  is,  the  rectangular  currents  in  successive  arma- 


234     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


7 


FIG.   130. — Alternating  current  and  direct  current  in  coil  between  d  and  a\  or 

a*  Fig.  127. 


FIG.  131. — Resultant  of  currents  given  in  Fig.  130. 


FIG.  132. — Alternating  current  and  direct  current  in  coil  between  d  and 
or  a2,  Fig.  127. 


FIG.  133. — Resultant  of  currents  shown  in  132. 


SYNCHRONOUS  CONVERTERS  235 

ture  coils  are  successively  displaced  in  phase  from  each  other; 
and  since  the  alternating  current  is  the  same  in  the  whole  section 
ai  a2,  and  in  phase  with  the  rectangular  current  in  the  coil  d, 
it  becomes  more  and  more  out  of  phase  with  the  rectangular 
current  when  passing  from  coil  d  toward  ai  or  a2,  as  shown  in 
Figs.  130  to  133,  until  the  maximum  phase  displacement  between 
alternating  and  rectangular  current  is  reached  at  the  alternating 

leads  ai  and  a2,  and  is  equal  to  -• 

li 

89.  Thus,  if  E  =  direct  voltage,  and  I  =  direct  current,  in 
an  armature  coil  displaced  by  angle  T  from  the  position  d,  mid- 
way between  two  adjacent  leads  of  the  n-phase  converter,  the 

direct  current  is  ~  for  the  half  period  from  0  to  ?r,  and  the  alter- 
nating current  is 

V2  I'  sin  (0  -  r), 
where 

/'  = 


n  sin  - 
n 

is  the  effective  value  of  the  alternating  current.     Thus,  the  actual 
current  in  this  armature  coil  is 

io  =  \/2  /'  sin  (0  —  T)  -  g 

_  /  [4  sin  (0-r)  _ 

"2 

nsin- 

(  n 

In  a  double-current  generator,  instead  of  the  minus  sign,  a 
plus  sign  would  connect  the  alternating  and  the  direct  current 
in  the  parenthesis. 

The  effective  value  of  the  resultant  converter  current  thus  is: 


n   sin  -  rnr  sin  - 

n  n 


Since  ~  is  the  current  in  the  armature  coil  of  a  direct-current 


236     ELEMENTS  OF  ELECTRICAL  ENGINEERING 
generator  of  the  same  output,  we  have 

7r   = 


Jo' 

2                    o 

|      i 

16  COS*T 

7 
2 

—                               1      -1 

n2  sin2  - 
n 

nir  sin  ^ 

71 

the  ratio  of  the  power  loss  in  the  armature  coil  resistance  of  the 
converter  to  that  of  the  direct-current  generator  of  the  same 
output,  and  thus  the  ratio  of  coil  heating. 

This  ratio  is  a  maximum  at  the  position  of  the  alternating 

leads,  T  =  -,  and  is 


7m    = 

n*  sin 


n 


It  is  a  minimum  for  a  coil  midway  between  adjacent  alter- 
nating leads,  T  =  0,  and  is 

=        8      ...  i 

.         7T  .IT 

n2  sin2  -  mr  sin  - 

n  n 

Integrating  over  T  from  0  (coil  d)  to-,  that  is,  over  the  whole 
phase  or  section  0,1  0,%,  we  have 


the  ratio  of  the  total  power  loss  in  the  armature  resistance  of 
an  n-phase  converter  to  that  of  the  same  machine  as  direct- 
current  generator  at  the  same  output,  or  the  relative  armature 
heating. 

Thus,  to  get  the  same  loss  in  the  armature  conductors,  and 
consequently  the  same  heating  of  the  armature,  the  current  in 
the  converter,  and  thus  its  output,  can  be  increased  in  the  pro- 
portion —7=  over  that  of  the  direct-current  generator. 

The  calculation  for  the  two-circuit  single-phase  converter  is 
somewhat  different,  since  in  this  in  one-third  of  the  armature 
the  Pr  loss  is  that  of  the  direct-current  output,  and  only  in  the 

27f 

other  two-thirds — or  an  arc  -^ is  there  alternating  current. 

o 


SYNCHRONOUS  CONVERTERS        237 

Thus  in  an  armature  coil  displaced  by  angle  r  from  the  center  of 
this  latter  section  the  resultant  current  is 

io  =  V2  /'  sin  (0  -  r)  - 


giving  the  effective  value 


I     III          16 

°W:= 

thus,  the  relative  heating  is 

//oV      11          16 

^  (I  = 

\2/ 
with  the  minimum  value  at  r  =  0,  it  is 

•"•  -  T  ~        =  °-70' 


and  with  the  maximum  value  at  r  =  ^  it  is 

o 

11          8 

=2-18; 


the  average  current  heating  in  two-thirds  of  the  armature  is 

11          48          TT 
•T  dT  =  -3-  -  ^—/^  Sln  3 


3  7T2 

in  the  remaining  third  of  the  armature,  Tz  =  1,  thus  the  average 
is 


3 

=  1.151, 

and  therefore  the  rating  is 

-4=  =  0.93. 

Vr 

By  substituting  for  n,  in  the  general  equations  of  current  heat- 
ing and  rating  based  thereon,  numerical  values,  we  get  the 
following  table: 


238     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


d 

V 

•si 

0> 

§ 

0) 

Type 

Direct-cur 
generator 

P 

jj-fl 

£•§ 

• 

a 

£ 

H 

a 

1 

1 

8 

n  

2 

2 

3 

4 

6 

12 

1.00 

0.45 

0.70 

0.225 

0.20 

0.19 

0.187 

7m  • 

r  

1.00 
1.00 

3.00 
1.37 

2.18 
1.157 

1.20 
0.555 

0.73 
0.37 

0.42 
0.26 

0.24 
0.20 

0.187 

Rating     (by 

mean    arm. 

heating) 

1.00 

0.85 

0.93 

1.34 

1.64 

1.96 

2.24 

2.31 

As  seen,  in  the  two-circuit  single-phase  converter  the  arma- 
ture heating  is  less,  and  more  uniformly  distributed,  than  in  the 
single-circuit  single-phase  converter. 

90.  A  very  great  gain  is  made  in  the  output  by  changing 
from  three-phase  to  six-phase,  but  relatively  little  by  still 
further  increasing  the  number  of  phases. 

In  these  values,  the  small  power  component  of  current  supply- 
ing the  losses  in  the  converter  has  been  neglected. 

These  values  apply  only  to  the  case  where  the  alternating 
current  is  in  phase  with  the  supply  voltage,  that  is,  for  unity 
power-factor  of  supply.  If,  however,  the  current  lags,  or  leads, 
by  the  time  angle  0,  then  the  alternating  current  and  direct 
current  are  not  in  opposition  in  the  armature  coil  d  midway 
between  adjacent  leads,  Fig.  127,  and  the  resultant  current  is 
a  minimum  and  of  the  shape  shown  in  Fig.  128,  at  a  point 
of  the  armature  winding  displaced  from  mid  position  d  by  angle 
r  =  0.  At  the  leads  the  displacement  between  alternating  cur- 


7T         •         7T 

rent  and  direct  current  then  is  not  -,  but  -  +  8     at     the 

n          n 


one, 


6  at  the  other  lead,  and  thus  at  the  other  side  of  the  same 

n 

lead.  The  resultant  current  is  thus  increased  at  the  one,  de- 
creased at  the  other  lead,  and  the  heating  changed  accordingly. 
For  instance,  in  a  quarter-phase  converter  at  zero  phase  dis- 
placement, the  resultant  current  at  the  lead  would  be  as  shown  in 

Fig.  134,  -  =  45  deg.,  while  at  30  deg.  lag  the  resultant  currents 
in  the  two  coils  adjacent  to  the  commutator  lead  are  displaced 


SYNCHRONOUS  CONVERTERS        239 

respectively  by-  +  &  =  75  deg.  and  by  -  —  d  =  15  deg.,  and  so 

of  very  different  shape,  as  shown  by  Figs.  135  and  136,  giving 
very  different  local  heating.  Phase  displacement  thus  increases 
the  heating  at  the  one,  decreases  it  at  the  other  side  of  each 
commutator  lead. 

Let  again, 

I  =  direct  current  per  commutator  brush. 

The  effective  value  of  the  alternating  power  current  in  the 
armature  winding,  or  ring  current,  corresponding  thereto,  is 


n  sn  - 
n 

Let  pi'  =  total  power  current,  'allowing  for  the  losses  of  power 
in  the  converter;  qlf  =  reactive  current  in  the  converter,  assumed 
as  positive  when  lagging,  as  negative  when  leading,  and  si'  = 
total  current,  where  s  =  Vp2  +  tf2  is  the  ratio  of  total  current  to 
the  load  current,  that  is,  power  current  corresponding  to  the 

direct-current  output,  and  —  =  tan   6  is  the   time  lag  of  the 

supply  current;  p  is  a  quantity  slightly  larger  than  1,  by  the 
losses  in  the  converter,  or  slightly  smaller  than  1  in  an  inverted 
converter. 

The  actual  current  in  an  armature  coil  displaced  in  position 
by  angle  r  from  the  middle  position  d  between  the  adjacent 
collector  leads,  then,  is 

to  =  V2  If  [p  sin  (0  -  T)  -  q  cos  (0  -  r)  }  -  ^ 

4 

[p  sin  (0  —  T)  —  q  cos  (0  —  T)]  —  I 


2  |  n  sin  - 
n 


and,  therefore,  its  effective  value  is 


=  \l  I  * 

*vjo 


/    I         8(p2  +  g2)        16  (p  COST  +  gsinr) 


n 


/                 8  s2  16  s  cos  (T  -  6) 

2 11  +  ~  rr^r  •  —^r 

n2  sin2  -  TTH  sin  - 

n  n 


240     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


\ 


FIG.  134. — Quarter-phase  converter  unity  power-factor,  armature  current 
at  collector  lead. 


\ 

\ 


v_ 


FIG.  135. — Quarter-phase  converter  phase  displacement  30 
ture  current  at  collector  lead. 


7 


FIG.  136. — Quarter-phase  converter  phase  displacement  30  degrees,  arma- 
ture current  at  collector  lead. 


SYNCHRONOUS  CONVERTERS  241 

and  herefrom  the  relative  heating  in  an  armature  coil  displaced 
by  angle  r  from  the  middle  between  adjacent  commutator  leads: 

8s2  16scos(r-0) 


n  sn  -  irn  sn  — 

n  n 


this  gives  at  the  leads,  or  for  r  =  +  —  > 


8s 


n2  sin2  —  TTU  sin  — 

n  n 

16  s  cos  (—  —  0) 
'8s2  \n        I 


n2  sin2  -  irn  sin  - 

n  n 

Averaging  from to  H gives  the  mean  current-heating  of 

Ti  ft 

the  converter  armature. 

r 


1  + 


J+^ 
%* 
-? 


n2  sin2  — 

n2  sin  - 

n 

n 

i    i 

8s2 

16s  cos  0 

—  i  -f- 

9 

*7T 

7T"2 

n2  sin2  — 

n 

-t-i_ 

8  (p2  +  q* 

)            16  P 

2      2  - 
n 


91.  This  gives  for 

Three-phase,  n  =  3: 

7T  =  1  +  1.185  s2  -  1.955  s  cos  (T  -  0), 
ym  =  1  +  1.185  s2  -  1.955  s  cos  (60  ±  0), 
r  =  1  +  1.185s2  -  1.620  p. 

Quarter-phase,  n  =  4: 

>YT  =  1  +  s2  -  1.795  s  cos  (r  -  0), 
ym  =  1  +  s2  -  1.795  s  cos  (45  ±  0), 
T  =  1  +  s2  -  1.620  ». 


242     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Six-phase,  n  =  6: 

7r  =  1  +  0.889  s2  -  1.695  s  cos  (r  -  0), 
7m  =  1  +  0.889  s2  -  1.695  s  cos  (30  +  0), 
r  =  1  +  0.889s2  -  1.62  p, 

oo  -phase,  n  =  co  : 

TT  =  7m  =  r  =  1  +  0.810  s2  -  1.62  s  cos  0 
=  1  +  0.810s2  -  1.62  p. 

Choosing  p  =  1.04,  that  is,  assuming  4  per  cent,  loss  in  friction 
and  windage,  core  loss  and  field  excitation — the  z'2r  loss  of  the 
armature  is  not  included  in  p,  as  it  is  represented  by  a  drop  of 
direct-current  voltage  below  that  corresponding  to  the  alternat- 
ing voltage,  and  not  by  an  increase  of  the  alternating  current 
over  that  corresponding  to  the  direct  current — we  get,  for  dif- 
ferent phase  angles  from  0  =  0  deg.  to  0  =  60  deg.,  the  values 
given  below: 

0=0  10        20      30        40        50        60 

s  =  -^          =1.04     1.0561.1081.20     1.36     1.62     2.08 
cos  0 

q  =  s  sin  0  = 

react'  cur>          =.     0        0.184  0.379  0.60    0.876  1.24     1.80 
power  cur. 

tan0  =0        0.176  0.364  0.577  0.839  1.192  1.732 

Three-phase: 


_ 

i    , 

1.62 

2 

08 

2.70 

3. 

65 

5. 

19 

8.16 

'*£  = 

*: 

26 

1.00 

0 

80 

0.68 

0. 

70 

0. 

99 

2.06 

r  = 

0. 

60 

0.64 

0 

77 

1.02 

1. 

51 

2. 

43 

4.45 

Quarter-phase  : 

7m    = 

1.02 

1 

39 

1.88 

2. 

64 

3. 

'87 

6.30 

7ft 

7m'    = 

• 

4  \J 

0.55 

0 

43 

0.38 

0. 

42 

0. 

73 

1.71 

r  =  - 

-  0. 

40 

0.43 

0 

54 

0.75 

1. 

16 

1. 

94 

3.64 

Six-phase  : 

7m    = 

\   o 

44 

0.62 

0 

88 

1.27 

1. 

86 

2. 

85 

4.85 

7m'    -      J 

1 

0.31 

0 

24 

0.25 

0. 

38 

0. 

75 

1.79 

r  = 

0. 

28 

0.31 

0 

41 

0.60 

0. 

97 

1. 

65 

3.17 

oo  -phase: 

-v    —  'v  '  —  r 

Jm            jm     ~       •*• 

=  0 

20 

0.22 

0 

32 

0.49 

0. 

82 

1. 

45 

2.82 

92.  The  values  are  shown  graphically  in  Figs.  137  and  138, 


SYNCHRONOUS  CONVERTERS 


243 


reactive  current         ,     . 
with  tan  6  =  -  TT~  as  abscissas,  and  7 


as  ordinates 


energy  current 
in  Fig.  137,  T  as  ordinates  in  Fig.  138. 

As  seen,  with  increasing  phase  displacement,  irrespectively 
whether  lag  or  lead,  the  average  as  well  as  the  maximum  arma- 
ture heating  very  greatly  increases.  This  shows  the  necessity 
of  keeping  the  power-factor  near  unity  at  full  load  and  overload, 
and  when  applied  to  phase  control  of  the  voltage  by  converter, 
means  that  the  shunt  field  of  the  converter  should  be  adjusted  so 
as  to  give  a  considerable  lagging  current  afho  load,  so  that  the 


~7 


7 


I 


7 


GENERATOR  HEA 


FIG.  137. — Maximum  72r  heating  in  converter  armature  coil  expressed  in  per 
cent,  of  direct-current  generator  72r  heating. 

current  comes  into  phase  with  the  voltage  at  about  full  load. 
It  therefore  is  very  objectionable  in  this  case  to  adjust  the  con- 
verter for  minimum  current  at  no  load,  as  occasionally  done  by 
ignorant  engineers,  since  such  wrong  adjustment  would  give  con- 
siderable leading  current  at  load,  and  therewith  unnecessary 
armature  heating. 

It  must  be  considered,  however,  that  above  values  are  referred 
to  the  direct-current  output,  and  with  increase  of  phase  angle 
the  alternating-current  input,  at  the  same  output,  increases, 


244     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

and  the  heating  increases  with  the  square  of  the  current.  Thus 
at  60  deg.  lag  or  lead,  the  power-factor  is  0.5,  and  the  alternating- 
current  input  thus  twice  as  great  as  at  unity  power-factor,  corre- 
sponding to  four  times  the  heating.  It  is  interesting  therefore 
to  refer  the  armature  heating  to  the  alternating-current  input, 
that  is,  compare  the  heating  of  the  converter  with  that  of  a 
synchronous  motor  of  the  same  alternating-current  input.  This 
is  given  by 

r      r 

1  1  —  ~S 


PER 

CENT 

60  / 

z 

/ 

320 

/ 

/ 

300/ 

/ 

/, 

/ 

/ 

260^ 

/ 

/ 

/ 

^ 

/ 

/ 

/ 

/ 

V 

/ 

/ 

/ 

200 

<% 

f 

// 

/ 

/ 
/ 

/ 

180 

A 

7 

3\ 

&< 

/ 

/ 

160 

/ 

^ 

'* 

y 

/ 

140 

/ 

i 

/ 

/ 

r 

& 

130 

D 

REG" 

CU 

iREf 

T 

/ 

/ 

/ 
/ 

// 

£\ 

100 

CEf 

ERA! 

OR 

HEAT 

ING, 

/^ 

/ 

s 
/ 

/" 

" 

80 

.  ' 

^ 

« 

X 

x 

60 

-—• 

^ 

^ 

x^ 

•^ 

40 


,  ' 

^ 

^^ 

F 

EAC1 

nvF 

CUR 

RFN- 

1  PF 

CF^ 

T 

?n 

1 

0       2 

D     3 

)      4 

OF 

0      5 

FUUL  UOAD  POWER  CU.RRE 
0      6,0     If)      80      90     100    1 

NT 

0    1 

0    1 

0    1 

0    1 

o 

FIG.  138.  —  Average  Izr  heating  in  converter  armature  expressed  in  per  cent. 
of  direct-current  generator  72r  heating. 

and,  for  p  =  1.04;  gives  the  following  values: 

0=  0  ^        10        20        30        40        50        60 

tan  (9  =  0          0.176  0.364  0.577  0.839  1.192  14.32 


Three-phase: 


Quarter-phase  : 


0.5550.57     0.63     0.71     0.82     0.93      1.03 


0.37     0.385  0.44    0.52     0.63     0.74      0.84 


SYNCHRONOUS  CONVERTERS 


245 


Six-phase: 

Ti  = 
oo -phase: 


0.26     0.28    0.335  0.42     0.52     0.63      0.73 
0.185  0.197  0.26    0.34    0.44    0.55      0.65 


It  is  seen  that,  compared  with  the  total  alternating-current 
input,  the  armature  heating  increases  much  less  with  increasing 
phase  displacement,  and  is  almost  always  much  lower  than  the 
heating  of  the  same  machine  at  the  same  input  and  phase  angle, 
when  running  a  synchronous  motor,  as  shown  in  Fig.  139. 


FIG.  139. — Average  I*r  heating  in  converter  armature  expressed  in  per 
cent,  of  synchronous  motor  72r  heating  at  the  same  power-factor. 

V.  Armature  Reaction 

93.  The  armature  reaction  of  the  polyphase  converter  is  the 
resultant  of  the  armature  reactions  of  the  machine  as  direct- 
current  generator  and  as  synchronous  motor.  If  the  com- 
mutator brushes  are  set  at  right  angles  to  the  field  poles  or 
without  lead  or  lag,  as  is  usually  done  in  converters,  the  direct- 
current  armature  reaction  consists  in  a  polarization  in  quadra- 
ture behind  the  field  magnetism.  The  armature  reaction  due 
to  the  power  component  of  the  alternating  current  in  a  synchro- 
nous motor  consists  of  a  polarization  in  quadrature  ahead  of 
the  field  magnetism,  which  is  opposite  to  the  armature  reaction 
as  direct-current  generator. 

Let  m  =  total  number  of  turns  on  the  bipolar  armature  or  per 
pair  of  poles  of  an  n-phase  converter,  /  =  direct  current,  then  the 

number  of  turns  in  series  between  the  brushes  =  -~,  hence  the 
total  armature  ampere-turns,  or  polarization,  =  -^—     Since,  how- 

16  * 


246     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


ever,  these  ampere-turns  are  not  unidirectional,  but  distributed 
over  the  whole  surface  of  the  armature,  their  resultant  is 


ml 


avg.  cos 


and,  since 


avg.  cos 


T        2 


ml 


we  have  F  =  — -  =  direct-current  polarization  of  the  converter 

7T 

(or  direct-current  generator)  armature. 

vn 

In  an  n-phase  converter  the  number  of  turns  per  phase  = 


n 


The  current  per  phase,  or  current  between  two  adjacent  leads 
(ring  current),  is 


7T 

n  sin  — 
n 


hence,  the  ampere-turn  per  phase, 

ml'       \/2  ml 


n 


n  sin  - 
n 


These  ampere-turns  are  distributed  over  -  of  the  circumference 
of  the  armature,  and  their  resultant  is  thus 

ml' 


and,  since 


we  have 


n 


avg.  cos 


avg.  cos 


n    .    TT 
=  -  sin  -i 

.  ;'»         n 


irn 


=  resultant  polarization, 


in  effective  ampere-turns  of  one  phase  of  the  converter. 

The  resultant  m.m.f.  of  n  equal  m.m.fs.  of  effective  value  of 
FI,  thus  maximum  value  of  FI  \/2,  acting  under  equal  angles 


SYNCHRONOUS  CONVERTERS 


247 


—  ,  and  displaced  in  phase  from  each  other  by  -  of  a  period,  or 

71  71 

phase  angle  —  ,  is  found  thus: 

=  Fi\/2sin  (0  --  —  j  =  one  of  the  m.m.fs.  of  phase 

2iir 
angle  0  =  --  ,  where  i  =  0,  1,  2  .    .   .  n  —  1,  acting  in  the  direc- 

tion T  =  -  ;  that  is.  the  zero  point  of  one  of  the  m.m.fs.  FI  is 

n  ' 

taken  as  zero  point  of  time  0,  and  the  direction  of  this  m.m.f. 
as  zero  point  of  direction  T. 

The  resultant  m.m.f.  in  any  direction  r  is  thus 


and,  since 


2  iir 


2   ir\         I          2  ir 

-  —  )COB(T-— 


we  have 


that  is,  the  resultant  m.m.f.  in  any  direction  T  has  the  phase 

6  =  r, 
and  the  intensity, 

rcFiA/2 
~^~ 

thus  revolves  in  space  with  uniform  velocity  and  constant  in- 
tensity, in  synchronism  with  the  frequency  of  the  alternating 
current. 


248     ELEMENTS  OF  ELECTRICAL  ENGINEERING 
Since  in  the  converter, 

Fl  =  ^M, 

TTU 

we  have 


the  resultant  m.m.f.  of  the  power  component  of  the  alternating 
current  in  the  n-phase  converter. 

This  m.m.f.  revolves  synchronously  in  the  armature  of  the 
converter;  and  since  the  armature  rotates  at  synchronism,  the 
resultant  m.m.f.  stands  still  in  space,  or,  with  regard  to  the  field 
poles,  in  opposition  to  the  direct-current  polarization.  Since 
it  is  equal  thereto,  it  follows  that  the  resultant  armature  reac- 
tions of  the  direct  current  and  of  the  corresponding  power 
component  of  the  alternating  current  in  the  synchronous  con- 
verter are  equal  and  opposite,  thus  neutralize  each  other,  and 
the  resultant  armature  polarization  equals  zero.  The  same  is 
obviously  the  case  in  an  inverted  converter,  that  is,  a  machine 
changing  from  direct  to  alternating  current. 

94.  The  conditions  in  a  single-phase  converter  are  different, 
however.  At  the  moment  when  the  alternating  current  =  0, 
the  full  direct-current  reaction  exists.  At  the  moment  when 
the  alternating  current  is  a  maximum,  the  reaction  is  the  differ- 
ence between  that  of  the  alternating  and  of  the  direct  current; 
and  since  the  maximum  alternating  current  in  the  single-phase 
converter  equals  twice  the  direct  current,  at  this  moment  the 
resultant  armature  reaction  is  equal  but  opposite  to  the  direct- 
current  reaction. 

Hence,  the  armature  reaction  oscillates  with  twice  the  fre- 
quency of  the  alternating  current,  and  with  full  intensity,  and 
since  it  is  in  quadrature  with  the  field  excitation,  tends  to  shift 
the  magnetic  flux  rapidly  across  the  field  poles,  and  thereby 
tends  to  cause  sparking  and  power  losses.  This  oscillating 
reaction  is,  however,  reduced  by  the  damping  effect  of  the  mag- 
netic field  structure.  It  is  somewhat  less  in  the  two-circuit 
single-phase  converter. 

Since  in  consequence  hereof  the  commutation  of  the  single- 
phase  converter  is  not  as  good  as  that  of  the  polyphase  con- 
verter, in  the  former  usually  voltage  commutation  has  to  be 
resorted  to;  that  is,  a  commutating  pole  used,  or  the  brushes 
shifted  from  the  position  midway  between  the  field  poles;  and 


SYNCHRONOUS  CONVERTERS        249 

in  the  latter  case  the  continuous-current  ampere-turns  inclosed 
by  twice  the  angle  of  lead  of  the  brushes  act  as  a  demagnetizing 
armature  reaction,  and  require  a  corresponding  increase  of  the 
field  excitation  under  load. 

While  the  absence  of  armature  reaction  eliminates  the  need  of 
a  commutating  pole  to  counteract  the  sparking  due  to  the  re- 
verse field  of  armature  reaction,  nevertheless,  commutating 
poles  are  very  often  used  in  converters,  to  control  the  high  self- 
induction  of  commutation,  which  economical  design  requires  in 
such  machines.  Such  commutating  poles  contain  only  the  am- 
pere turn  required  to  produce  the  commutating  flux,  thus  less 
than  in  generators. 

95.  Since  the  resultant  main  armature  reactions  neutralize 
each  other  in  the  polyphase  converter,  there  remain  only — 

1.  The  armature  reaction  due  to  the  small  power  component 
of  current  required  to  rotate  the  machine,  that  is,  to  cover  the 
internal  losses  of  power,  which  is  in  quadrature  with  the  field 
excitation  or  distorting,  but  of  negligible  magnitude. 

2.  The  armature  reaction  due  to  the  wattless  component  of 
alternating  current  where  such  exists. 

3.  An  effect  of  oscillating  nature,   which  may  be  called  a 
higher  harmonic  of  armature  reaction. 

The  direct  current,  as  rectangular  alternating  current  in  the 
armature,  changes  in  phase  from  coil  to  coil,  while  the  alternating 
current  is  the  same  in  a  whole  section  of  the  armature  between 
adjacent  leads. 

Thus  while  the  resultant  reactions  neutralize,  a  local  effect 
remains  which  in  its  relation  to  the  magnetic  field  oscillates 
with  a  period  equal  to  the  time  of  motion  of  the  armature  through 
the  angle  between  adjacent  alternating  leads;  that  is,  double 
frequency  in  a  single-phase  converter  (in  which  it  is  equal  in 
magnitude  to  the  direct-current  reaction,  and  is  the  oscillating 
armature  reaction  discussed  above),  sextuple  frequency  in  a 
three-phase  converter,  and  quadruple  frequency  in  a  four- 
phase  converter. 

The  amplitude  of  this  oscillation  in  a  polyphase  converter  is 
small,  arid  its  influence  upon  the  magnetic  field  is  usually  neg- 
ligible, due  to  the  damping  effect  of  the  field  spools,  which  act 
like  a  short-circuited  winding  for  an  oscillation  of  magnetism. 

A  polyphase  converter  on  unbalanced  circuit  can  be  con- 
sidered as  a  combination  of  a  balanced  polyphase  and  a  single- 
phase  converter;  and  since  even  single-phase  converters  operate 
quite  satisfactorily,  the  effect  of  unbalanced  circuits  on  the 


250     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

polyphase  converter  is  comparatively  small,  within  reasonable 
limits. 

Since  the  armature  reaction  of  the  direct  current  and  of  the 
alternating  current  in  the  converter  neutralize  each  other,  no 
change  of  field  excitation  is  required  in  the  converter  with  changes 
of  load. 

Furthermore,  while  in  a  direct-current  generator  the  arma- 
ture reaction  at  given  field  strength  is  limited  by  the  distortion 
of  the  field  caused  thereby,  this  limitation  does  not  exist  in  a 
converter;  and  a  much  greater  armature  reaction  can  be  safely 
used  in  converters  than  in  direct-current  generators,  the  dis- 
tortion being  absent  in  the  former. 

The  practical  limit  of  overload  capacity  of  a  converter  is  usu- 
ally far  higher  than  in  a  direct-current  generator,  since  the  arma- 
ture heating  is  relatively  small,  and  since  the  distortion  of  field, 
which  causes  sparking  on  the  commutator  under  overloads  in  a 
direct-current  generator,  is  absent  in  a  converter. 

The  theoretical  limit  of  overload — that  is,  the  overload  at 
which  the  converter  as  synchronous  motor  drops  out  of  step 
and  comes  to  a  standstill — is  usually  far  beyond  reach  at  steady 
frequency  and  constant  impressed  alternating  voltage,  while  on 
an  alternating  circuit  of  pulsating  frequency  or  drooping  voltage 
it  obviously  depends  upon  the  amplitude  and  period  of  the 
pulsation  of  frequency  or  on  the  drop  of  voltage. 

VI.  Reactive  Currents  and  Compounding 

96.  Since  the  polarization  due  to  the  power  component  of 
the  alternating  current  as  synchronous  motor  is  in  quadrature 
ahead  of  the  field  magnetization,  the  polarization  or  magnetizing 
effect  of  the  lagging  component  of  alternating  current  is  in 
phase,  that  of  the  leading  component  of  alternating  current  in 
oppositon  to  the  field  magnetization;  that  is,  in  the  converter 
no  magnetic  distortion  exists,  and  no  armature  reaction  at  all 
if  the  current  is  in  phase  with  the  impressed  e.m.f.,  while  the- 
armature  reaction  is  demagnetizing  with  a  leading  and  mag- 
netizing with  a  lagging  current. 

Thus  if  the  alternating  Current  is  lagging,  the  field  excitation 
at  the  same  impressed  e.m.f.  has  to  be  lower,  and  if  the  alter- 
nating current  is  leading,  the  field  excitation  has  to  be  higher, 
than  required  with  the  alternating  current  in  phase  with  the 


SYNCHRONOUS  CONVERTERS        251 

e.m.f.  Inversely,  by  raising  the  field  excitation  a  leading 
current,  or  by  lowering  it  a  lagging  current,  can  be  produced  in 
a  converter  (and  in  a  synchronous  motor). 

Since  the  alternating  current  can  be  made  magnetizing  or 
demagnetizing  according  to  the  field  excitation,  at  constant 
impressed  alternating  voltage,  the  field  excitation  of  the  con- 
verter can  be  varied  through  a  wide  range  without  noticeably 
affecting  the  voltage  at  the  commutator  brushes;  and  in  con- 
verters of  high  armature  reaction  and  relatively  weak  field,  full 
load  and  overload  can  be  carried  by  the  machine  without  any 
field  excitation  whatever,  that  is,  by  exciting  the  field  by  armature 
reaction  by  the  lagging  alternating  current.  Such  converters 
without  field  excitation,  or  reaction  converters,  must  always  run 
with  more  or  less  lagging  current,  that  is,  give  the  same  reaction 
on  the  line  as  induction  motors,  which,  as  known,  are  far  more 
objectionable  than  synchronous  motors  in  their  reaction  on  the 
alternating  system,  and  therefore  they  are  no  longer  used. 

Conversely,  however,  at  constant  impressed  alternating  vol- 
tage the  direct-current  voltage  of  a  converter  cannot  be  varied 
by  varying  the  field  excitation  (except  by  the  very  small  amount 
due  to  the  change  of  the  ratio  of  conversion),  but  a  change  of 
field  excitation  merely  produces  wattless  currents,  lagging  or 
magnetizing  with  a  decrease,  leading  or  demagnetizing  with  an 
increase  of  field  excitation.  Thus  to  vary  the  continuous- 
current  voltage  of  a  converter  usually  the  impressed  alternating 
voltage  has  to  be  varied.  This  can  be  done  either  by  potential 
regulator  or  compensator,  that  is,  transformers  of  variable  ratio 
of  transformation,  or  by  a  synchronous  machine  of  the  same 
number  of  poles  as  the  converter,  on  the  same  shaft  and  con- 
nected in  series  ("synchronous  booster")  or  by  the  effect  of  watt- 
less currents  on  self-inductance.  The  latter  method  is  especially 
suited  for  converters,  due  to  their  ability  of  producing  wattless 
currents  by  change  of  .field  excitation. 

The  e.m.f.  of  self -inductance  lags  90  deg.  behind  the  current; 
thus,  if  the  current  is  lagging  90  deg.  behind  the  impressed  e.m.f., 
the  e.m.f.  of  self-inductance  is  180  deg.  behind,  or  in  opposition 
to,  the  impressed  e.m.f.,  and  thus  reduces  it.  If  the  current  is 
90  deg.  ahead  of  the  e.m.f.,  the  e.m.f.  of  self-inductance  is  in 
phase  with  the  impressed  e.m.f.,  thus  adds  itself  thereto  and 
raises  it.  Therefore,  if  self-inductance  is  inserted  into  the  lines 
between  converter  and  constant-potential  generator,  and  a  watt- 


I 

252     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

less  lagging  current  is  produced  by  the  converter  by  a  decrease 
of  its  field  excitation,  the  e.m.f.  of  self-inductance  of  this  lagging 
current  in  the  line  lowers  the  alternating  impressed  voltage  at 
the  converter  and  thus  its  direct-current  voltage;  and  if  a  watt- 
less leading  current  is  produced  by  the  converter  by  an  increase 
of  its  field  excitation,  the  e.rn.f.  of  self-inductance  of  this  leading 
current  raises  the  impressed  alternating  voltage  at  the  converter 
and  thus  its  direct-current  voltage. 

97.  In  this  manner,  by  self-inductance  in  the  lines  leading  to 
the  converter,  its  voltage  can  be  varied  by  a  change  of  field 
excitation,   or  conversely  its  voltage   maintained   constant  at 
constant  generator  voltage  or  even  constant  generator  excita- 
tion, with  increasing  load  and  thus  increasing  resistance  drop 
in  the  line;  or  the  voltage  can  even  be  increased  with  increasing 
load,  that  is,  the  system  over-compounded. 

The  change  of  field  excitation  of  the  converter  with  changes 
of  load  can  be  made  automatic  by  the  combination  of  shunt  and 
series  field,  and  in  this  manner  a  converter  can  be  compounded 
or  even  over-compounded  similarly  to  a  direct-current  generator. 
While  the  effect  is  the  same,  the  action,  however,  is  different; 
and  the  compounding  takes  place  not  in  the  machine  as  with  a 
direct-current  generator,  but  in  the  alternating  lines  leading  to 
the  machine,  in  which  self-inductance  becomes  essential. 

As  the  reactance  of  the  transmission  line  is  rarely  sufficient 
to  give  phase  control  over  a  wide  range  without  excessive  reac- 
tive currents,  it  is  customary,  especially  at  25  cycles,  to  insert 
reactive  coils  into  the  leads  between  the  converter  and  its  step- 
down  transformers,  in  those  cases  in  which  automatic  phase 
control  by  converter  series  fields  is  desired,  as  in  power  trans- 
mission for  suburban  and  interurban  railways,  etc.,  or  to  specially 
design  the  step-down  transformers  for  high  internal  reactance. 
Usually  these  reactive  coils  are  designed  to  give  at  full-load 
current  a  reactance  voltage  equal  to  about  15  per  cent,  of  the 
converter  supply  voltage,  and  therefore  capable  of  taking  care 
of  about  10  per  cent,  line  drop  at  good  power-factors. 

VII.  Variable  Ratio  Converters   ("Split  Pole"  Converters) 

98.  With  a  sine  wave  of  alternating  voltage,  and  the  com- 
mutator brushes  set  at  the  magnetic  neutral,  that  is,  at  right 
angles  to  the  resultant  magnetic  flux,  the  direct  voltage  of  a 


SYNCHRONOUS  CONVERTERS        253 

converter  is  constant  at  constant  impressed  alternating  voltage. 
It  equals  the  maximum  value  of  the  alternating  voltage  between 
two  diametrically  opposite  points  of  the  commutator,  or  "dia- 
metrical voltage,"  and  the  diametrical  voltage  is  twice  the  voltage 
between  alternating  lead  and  neutral,  or  star  or  Y  voltage  of 
the  polyphase  system. 

A  change  of  the  direct  voltage,  at  constant  impressed  alter- 
nating voltage,  can  be  produced — 

Either  by  changing  the  position  angle  between  the  commu- 
tator brushes  and  the  resultant  magnetic  flux,  so  that  the  direct 
voltage  between  the  brushes  is  not  the  maximum  diametrical 
alternating  voltage  but  only  a  part  thereof, 

Or  by  changing  the  maximum  diametrical  alternating  voltage, 
at  constant  effective  impressed  voltage,  by  wave-shape  distortion 
by  the  superposition  of  higher  harmonics. 

In  the  former  case,  only  a  reduction  of  the  direct  voltage 
below  the  normal  value  can  be  produced,  while  in  the  latter 
case  an  increase  as  well  as  a  reduction  can  be  produced,  an 
increase  if  the  higher  harmonics  are  in  phase,  and  a  reduction 
if  the  higher  harmonics  are  in  opposition  to  the  fundamental 
wave  of  the  diametrical  or  Y  voltage. 

Both  methods  are  combined  in  the  so-called  "  Regulating  Pole 
Converter"  or  "Split  Pole  Converter,"  which  is  used  to  supply, 
from  constant  alternating  voltage  supply,  direct  voltage  varying 
sometimes  over  a  range  of  ±  20  per  cent. 

In  this  type  of  converter,  the  field  pole  is  divided  into  sections, 
usually  two,  a  smaller  one,  the  regulating  pole,  and  a  larger  one, 
the  main  pole.  By  varying  the  excitation  of  the  regulating  pole 
from  maximum  in  one  direction,  to  maximum  in  the  opposite 
direction,  the  direction  of  the  resultant  magnetic  field  flux,  and 
the  effective  width  of  the  field  pole,  and  with  the  latter  the 
wave  shape,  are  varied.  To  keep  the  wave  shape  variation 
local  in  the  converter,  so  as  not  to  reflect  it  into  the  primary 
supply  circuit,  the  proper  transformer  connection  must  be 
used.  This  is  Y  primary  with  preferably  A  or  double  delta 
(for  three-phase  and  for  six-phase)  or  Y  and  double  Y  or  dia- 
metrical in  the  secondary. 

Vm.     Starting 

99.  The  polyphase  converter  is  self-starting  from  rest;  that 
is,  when  connected  across  the  polyphase  circuit  it  starts,  acceler- 


254     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

ates,  and  runs  up  to  complete  synchronism.  The  e.m.f.  between 
the  commutator  brushes  is  alternating  in  starting,  with  the  fre- 
quency of  slip  below  synchronism.  Thus  a  direct-current  volt- 
meter or  incandescent  lamps  connected  across  the  commutator 
brushes  indicate  by  their  beats  the  approach  of  the  converter  to 
synchronism.  When  starting,  the  field  circuit  of  the  converter 
has  to  be  opened  or  at  least  greatly  weakened.  The  starting 
of  the  polyphase  converter  is  largely  a  hysteresis  effect  and 
entirely  so  in  machines  with  laminated  field  poles,  while  in  ma- 
chines with  solid  magnet  poles  or  with  a  short-circuited  winding 
(squirrel-cage)  in  the  field  poles,  secondary  currents  in  the  latter 
contribute  to  the  starting  torque,  but  at  the  same  time  reduce 
the  magnetic  starting  flux  by  their  demagnetizing  effect.  The 
torque  is  produced  by  the  attraction  between  the  alternating 
currents  of  the  successive  phases  upon  the  remanent  magnetism 
and  secondary  currents  produced  by  the  preceding  phase.  It 
is  necessarily  comparatively  weak,  and  from  full-load  to  twice 
full-load  current  at  from  one-third  to  one-half  of  full  voltage  is 
required  to  start  from  rest  without  load.  Usually,  low-voltage 
taps  on  the  transformers  are  used  to  give  the  lower  starting 
voltage. 

While  an  induction  motor  can  never  reach  exact  synchronism, 
but  must  even  at  no  load  slip  slightly  to  produce  the  friction 
torque,  the  converter  or  synchronous  motor  reaches  exact  syn- 
chronism, due  to  the  difference  of  the  magnetic  reluctance  in  the 
direction  of  the  field  poles  and  in  the  direction  in  electrical 
quadrature  thereto;  that  is,  the  field  structure  acts  like  a  shuttle 
armature  and  the  polar  projections  catch  with  the  rotating 
magnet  poles  in  the  armature,  in  a  similar  way  as  an  induction 
motor  armature  with  a  single  short-circuited  coil  (synchronous 
induction  motor,  reaction  machine)  drops  into  step.  Obviously, 
the  single-phase  converter  is  not  self-starting. 

At  the  moment  of  starting,  the  field  circuit  of  the  converter  is 
in  the  position  of  a  secondary  to  the  armature  circuit  as  primary; 
and  since  in  general  the  number  of  field  turns  is  very  much  larger 
than  the  number  of  armature  turns,  excessive  e.m.fs.  may  be 
generated  in  the  field  circuit,  reaching  frequently  4000  to  6000 
volts,  which  have  to  be  taken  care  of  by  some  means,  as  by 
breaking  the  field  circuit  into  sections,  or  protecting  against  ex- 
cessive voltages  by  a  squirrel-cage  starting  winding  in  the  pole 
faces.  As  soon  as  synchronism  is  reached,  which  usually  takes 


SYNCHRONOUS  CONVERTERS        255 

from  a  few  seconds  to  a  minute  or  more,  and  is  seen  by  the  ap- 
pearance of  continuous  voltage  at  the  commutator  brushes,  the 
field  circuit  is  closed  and  the  load  put  on  the  converter.  Ob- 
viously, while  starting,  the  direct-current  side  of  the  converter 
must  be  open-circuited,  since  the  e.m.f.  between  commutator 
brushes  is  alternating  until  synchronism  is  reached. 

When  starting  from  the  alternating  side,  the  converter  can 
drop  into  synchronism  at  either  polarity;  but  its  polarity  can  be 
reversed  by  strongly  exciting  the  field  in  the  right  direction  by 
some  outside  source,  as  another  converter,  etc.,  or  by  momen- 
tarily opening  the  circuit  and  thereby  letting  the  converter  slip 
one  pole. 

Since  when  starting  from  the  alternating  side  the  converter 
requires  a  very  large  and,  at  the  same  time,  lagging  current,  it 
is  occasionally  preferable  to  start  it  from  the  direct-current  side 
as  direct-current  motor.  This  can  be  done  when  connected  to 
storage  battery  or  direct-current  generator.  When  feeding  into 
a  direct-current  system  together  with  other  converters  or  con- 
verter stations,  all  but  the  first  converter  can  be  started  from 
the  continuous  current  side  by  means  of  rheostats  inserted  into 
the  armature  circuit. 

To  avoid  the  necessity  of  synchronizing  the  converter,  by  phase 
lamps,  with  the  alternating  system  in  case  of  starting  by  direct 
current  (which  operation  may  be  difficult  where  the  direct 
voltage  fluctuates,  owing  to  heavy  fluctuations  of  load,  as  rail- 
way systems),  it  is  frequently  preferable  to  run  the  converter 
up  to  or  beyond  synchronism  by  direct  current,  then  cut  off 
from  the  direct  current,  open  the  field  circuit  and  connect  it  to 
the  alternating  system,  thus  bringing  it  into  step  by  alternating 
current. 

If  starting  from  the  alternating  side  is  to  be  avoided,  and 
direct  current  not  always  available,  as  when  starting  the  first 
converter,  a  small  induction  motor  (of  less  poles  than  the  con- 
verter) is  used  as  starting  motor. 

Converters  usually  are  started  from  the  alternating  side. 

IX.  Inverted  Converters 

100. .  Converters  may  be  used  to  change  either  from  alter- 
nating to  direct  current  or  as  inverted  converters  from  direct  to 
alternating  current.  While  the  former  use  is  by  far  the  more 


256     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

frequent,  sometimes  inverted  converters  are  desirable.  Thus  in 
low-tension  direct-current  systems  outlying  districts  have  been 
supplied  by  converting  from  direct  to  alternating,  transmitting 
as  alternating,  and  then  reconverting  to  direct  current.  Or  in 
a  station  containing  direct-current  generators  for  short-distance 
supply  and  alternators  for  long-distance  supply,  the  converter 
may  be  used  as  the  connecting  link  to  shift  the  load  from  the 
direct  to  the  alternating  generators,  or  inversely,  and  thus  be 
operated  either  way  according  to  the  distribution  of  load  on  the 
system.  Or  inverted  operation  may  be  used  in  emergencies  to 
produce  alternating  current. 

When  converting  from  alternating  to  direct  current,  the  speed 
of  the  converter  is  rigidly  fixed  by  the  frequency,  and  cannot  be 
varied  by  its  field  excitation,  the  variation  of  the  latter  merely 
changing  the  phase  relation  of  the  alternating  current.  When 
converting,  however,  from  direct  to  alternating  current  as  the 
only  source  of  alternating  current,  that  is,  not  running  in  multiple 
with  engine-  or  turbine-driven  alternating-current  generators,  the 
speed  of  the  converter  as  direct-current  motor  depends  upon  the 
field  strength;  thus  it  increases  with  decreasing  and  decreases 
with  increasing  field  strength.  As  alternating-current  generator, 
however,  the  field  strength  depends  upon  the  intensity  and 
phase  relation  of  the  alternating  current,  lagging  current  reducing 
the  field  strength  and  thus  increasing  speed  and  frequency,  and 
leading  current  increasing  the  field  strength  and  thus  decreasing 
speed  and  frequency. 

Thus,  if  a  load  of  lagging  current  is  put  on  an  inverted  con- 
verter, as,  for  instance,  by  starting  an  induction  motor  or  another 
converter  thereby  from  the  alternating  side,  the  demagnetizing 
effect  of  the  alternating  current  reduces  the  field  strength  and 
causes  the  converter  to  increase  in  speed  and  frequency.  An  in- 
crease of  frequency,  however,  may  increase  the  lag  of  the  current, 
and  thus  its  demagnetizing  effect,  and  thereby  still  further  in- 
crease the  speed,  so  that  the  acceleration  may  become  so  rapid  as 
to  be  beyond  control  by  the  field  rheostat  and  endanger  the 
machine.  Hence  inverted  converters  have  to  be  carefully 
watched,  especially  when  starting  other  converters  from  them, 
and  some  absolutely  positive  device  is  necessary  to  cut  the  in- 
verted converter  off  the  circuit  entirely  as  soon  as  its  speed  ex- 
ceeds the  danger  limit.  The  relatively  safest  arrangement  is 
separate  excitation  of  the  inverted  converter  by  an  exciter 


SYNCHRONOUS  CONVERTERS        257 

mechanically  driven  thereby,  since  an  increase  of  speed  in- 
creases the  exciter  voltage  at  a  still  higher  rate,  and  thereby  the 
excitation  of  the  converter,  and  thus  tends  to  check  its  speed. 

This  danger  of  racing  does  not  exist  if  the  inverted  converter 
operates  in  parallel  with  alternating  generators,  provided  that 
the  latter  and  their  prime  movers  are  of  such  size  that  they 
cannot  be  carried  away  in  speed  by  the  converter.  In  an  in- 
verted converter  running  in  parallel  with  alternators  the  speed  is 
not  changed  by  the  field  excitation,  but  a  change  of  the  latter 
merely  changes  the  phase  relation  of  the  alternating  current 
supplied  by  the  converter;  that  is,  the  converter  receives  power 
from  the  direct-current  system,  and  supplies  power  into  the  alter- 
nating-current system  but  at  the  same  time  receives  wattless 
current  from  the  alternating  system,  lagging  at  under-excitation, 
leading  at  over-excitation,  and  can  in  the  same  way  as  an  ordinary 
converter  or  synchronous  motor  be  used  to  compensate  for  watt- 
less currents  in  other  parts  of  the  alternating  system,  or  to  regu- 
late the  voltage  by  phase  control. 

X.  Frequency 

101.  While  converters  can  be  designed  for  any  frequency,  the 
use  of  high  frequency,  as  60  cycles,  imposes  more  severe  limita- 
tions on  the  design,  especially  that  of  the  commutator,  as  to 
make  the  high-frequency  converter  inferior  to  the  low-frequency 
or  25-cycle  converter. 

The  commutator  surface  moves  the  distance  from  brush  to 
next  brush,  or  the  commutator  pitch,  during  one-half  cycle, 
that  is,  3^50  second  with  a  25-cycle,  J^20  second  with  a  60-cycle 
converter.  The  peripheral  speed  of  the  commutator,  however, 
is  limited  by  mechanical,  electrical,  and  thermal  considera- 
tions— centrifugal  forces,  loss  of  power  by  brush  friction,  and 
heating  caused  thereby.  The  limitation  of  peripheral  speed 
limits  the  commutator  pitch.  Within  this  pitch  must  be  in- 
cluded as  many  commutator  segments  as  necessary  to  take  care 
of  the  voltage  from  brush  to  brush,  and  these  segments  must  have 
a  width  sufficient  for  mechanical  strength.  With  the  smaller 
pitch  required  for  high  frequency,  this  may  become  impossible, 
and  the  limits  of  conservative  design  thus  may  have  to  be 
exceeded. 

In  a  converter,  due  to  the  absence  of  armature  reaction  and 
field  distortion,  a  higher  voltage  per  commutator  segment  can  be 


258     ELEMENTS  OF  ELECTRICAL  ENGINEERING      . 

allowed  than  in  a  direct-current  generator.  Assuming  17  volts 
as  limit  of  conservative  design  would  give  for  a  600-volt  con- 
verter 36  segments  from  brush  to  brush.  Allowing  0.2  inch 
for  segment  and  insulation,  as  minimum  conservative  value,  37 
segments  give  a  pitch  of  7.4  inches.  Estimating  5000  feet  per 
minute  as  conservative  limit  of  commutator  speed  gives  83.3 
feet  or  1000  inches  peripheral  speed  per  second,  and  with  7.4 
inches  pitch  this  gives  136  half  cycles,  or  68  cycles,  as  limit 
of  the  frequency,  permitting  conservative  commutator  design. 

At  60  cycles  higher  voltage  per  segment,  narrower  segments 
and  higher  commutator  speeds  thus  are  necessary  than  at  25 
cycles,  and  the  60-cycle  converter,  though  still  within  conserva- 
tive limits,  does  not  permit  as  conservative  commutator  design, 
especially  at  higher  voltage,  as  a  low-frequency  converter,  and 
a  lower  self-inductance  of  commutation  thus  must  be  aimed  at 
than  permissible  in  a  25-cycle  converter,  the  more  so  as  the  fre- 
quency of  commutation  (half  the  number  of  commutator  seg- 
ments per  pole  times  frequency  of  rotation)  necessarily  is  higher 
in  the  60-cycle  converter.  .  . 

Somewhat  similar  considerations  also  apply  to  the  armature 
construction :  the  peripheral  speed  of  the  armature,  even  if  chosen 
higher  for  the  60-cycle  converter,  limits  the  pitch  per  pole  at  the 
armature  circumference,  and  thereby  the  ampere  conductors 
per  pole  and  thus  the  armature  reaction,  the  more  so  as  shallower 
slots  are  necessary.  The  60-cycle  converter  cannot  be  built  with 
anything  like  the  same  armature  reaction  as  is  feasible  at  lower 
frequency.  On  the  armature  reaction,  however,  very  largely 
depends  the  stability  of  a  synchronous  motor  or  converter,  and 
machines  of  low  armature  reaction  tend  far  more  to  surging 
and  pulsation  of  current  and  voltage  than  machines  of  high 
armature  reaction. 

The  60-cycle  converter  therefore  cannot  be  made  quite  as 
stable  and  capable  of  taking  care  of  violent  fluctuations  of  load 
and  of  excessive  overloads  as  25-cycle  converters  can,  and  in 
this  respect  the  lower-frequency  machine  is  preferable,  though 
under  reasonably  favorable  conditions  regarding  variations  of 
load,  variations  of  supply  voltage,  and  overload  60-cycle  con- 
verters give  excellent  service. 

It  is  this  inherent  inferiority  of  the  60-cycle  converter  which 
has  largely  been  instrumental  in  introducing  25  cycles  as  the 
frequency  of  electric  power  generation  and  distribution. 


SYNCHRONOUS  CONVERTERS        259 

At  25  cycles,  converters  are  used  on  railway  load — the  most 
fluctuating  and  therefore  most  severe  service — built  for  1200 
volts,  and  even  still  much  higher  voltages  are  available. 

XI.  Double-current  Generators 

102.  Similar  in  appearance  to  the  converter,  which  changes 
from  alternating  to  direct  current,  and  to  the  inverted  converter, 
which  changes  from  direct  to  alternating  current,  is  the  double- 
current  generator;  that  is,  a  machine  driven  by  mechanical  power 
and  producing  direct  current  as  well  as  alternating  current  from 
the  same  armature,  which  is  connected  to  commutator  and  col- 
lector rings  in  the  same  way  as  in  the  converter.  Obviously  the 
use  of  the  double-current  generator  is  limited  to  those  sizes  and 
speeds  at  which  a  good  direct-current  generator  can  be  built  with 
the  same  number  of  poles  as  a  good  alternator,  that  is,  low- 
frequency  machines  of  large  output  and  relatively  high  speed; 
while  high-frequency  low-speed  double-current  generators  are 
undesirable. 

The  essential  difference  between  double-current  generator  and 
converter  is,  however,  that  in  the  former  the  direct  current  and 
the  alternating  current  are  not  in  opposition  as  in  the  latter,  but 
in  the  same  direction,  and  the  resultant  armature  polarization 
thus  the  sum  of  the  armature  polarization  of  the  direct  current 
and  of  the  alternating  current. 

Since  at  the  same  output  and  the  same  field  strength  the  arma- 
ture polarization  of  the  direct  current  and  that  of  the  alternating 
current  are  the  same,  it  follows  that  the  resultant  armature  polari- 
zation of  the  double-current  generator  is  proportional  to  the  load 
regardless  of  the  proportion  in  which  this  load  is  distributed 
between  the  alternating-  and  direct-current  sides.  The  heating  of 
the  armature  due  to  its  resistance  depends  upon  the  sum  of  the 
two  currents,  that  is,  upon  the  total  load  on  the  machine.  Hence, 
the  output  of  the  double-current  generator  is  limited  by  the 
current  heating  of  the  armature  and  by  the  field  distortion  due 
to  the  armature  reaction,  in  the  same  way  as  in  a  direct-current 
generator  or  alternator,  and  is  consequently  much  less  than  that 
of  a  converter. 

In  double-current  generators,  owing  to  the  existence  of  arma- 
ture reaction  and  consequent  field  distortion,  the  commutator 
brushes  are  more  or  less  shifted  against  the  neutral,  and  the 


260     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

direction  of  the  continuous-current  armature  polarization  is  thus 
shifted  against  the  neutral  by  the  same  angle  as  the  brushes. 
The  direction  of  the  alternating-current  armature  polarization, 
however,  is  shifted  against  the  neutral  by  the  angle  of  phase 
displacement  of  the  alternating  current.  In  consequence  thereof, 
the  reactions  upon  the  field  of  the  two  parts  of  the  armature  polari- 
zation, that  due  to  the  continuous  current  and  that  due  to  the 
alternating  current,  are  usually  different.  The  reaction  on  the 
field  of  the  direct-current  load  can  be  overcome  by  a  series  field. 
The  reaction  on  the  field  of  the  alternating-current  load  when 
feeding  converters  can  be  compensated  for  by  a  change  of  phase 
relation,  by  means  of  a  series  field  on  the  converter,  with  self- 
inductance  in  the  alternating  lines,  or  reactive  coils  at  the 
converters. 

Thus,  a  double-current  generator  feeding  on  the  alternating 
side  converters  can  be  considered  as  a  direct-current  generator  in 
which  a  part  of  the  commutator,  with  a  corresponding  part  of  the 
series  field,  is  separated  from  the  generator  and  located  at  a 
distance,  connected  by  alternating  leads  to  the  generator.  Ob- 
viously, automatic  compounding  of  a  double-current  generator  is 
feasible  only  if  the  phase  relation  of  the  alternating  current 
changes  from  lag  at  no  load  to  lead  at  load,  in  the  same  way  as 
produced  by  a  compounded  converter.  Otherwise,  rheostatic 
control  of  the  generator  is  necessary.  This  is,  for  instance,  the 
case  if  the  voltage  of  the  double-current  generator  has  to  be  varied 
to  suit  the  conditions  of  its  direct-current  load,  and  the  voltage 
of  the  converter  at  the  end  of  the  alternating  lines  varied  to  suit 
the  conditions  of  load  at  the  receiving  end,  independent  of  the 
voltage  at  the  double-current  generator,  by  means  of  alternating 
"potential  regulators  or  compensators. 

Compared  with  the  direct-current  generator,  the  field  of  the 
double-current  generator  must  be  such  as  to  give  a  much  greater 
stability  of  voltage,  owing  to  the  strong  demagnetizing  effect 
which  may  be  exerted  by  lagging  currents  on  the  alternating  side, 
and  may  cause  the  machine  to  lose  its  excitation  altogether. 
For  this  reason  it  is  frequently  preferable  to  excite  double-current 
generators  separately.  With  the  general  adoption  of  large 
three-phase  steam-turbine  units  for  electric  power  generation,  the 
use  of  inverted  converter  and  double-current  generator  has  greatly 
decreased. 


SYNCHRONOUS  CONVERTERS        261 

XII.  Conclusion 

103.  Of  the  types  of  machines,  converter,  inverted  converter, 
and  double-current  generator,  sundry  combinations  can  be  de- 
devised  with  each  other  and  with  synchronous  motors,  alternators, 
direct-current  motors  and  generators.     Thus,   for  instance,   a 
converter  can  be  used  to  supply  a  certain  amount  of  mechanical 
power   as   synchronous   motor.     In   this   case   the   alternating 
current  is  increased  beyond  the  value  corresponding  to  the  direct 
current  by  the  amount  of  current  giving  the  mechanical  power, 
and  the  armature  reactions  do  not  neutralize  each  other,  but 
the  reaction  of  the  alternating  current  exceeds  that  of  the  direct 
current  by  the  amount  corresponding  to  the  mechanical  load. 
In  the  same  way  the  current  heating  of  the  armature  is  in- 
creased.    An  inverted  converter  can  also  be  used  to  supply 
some  mechanical  power.     Either  arrangement,  however,  while 
quite  feasible,  has  the  disadvantage  of  interfering  with  auto- 
matic control  of  voltage  by  compounding. 

Double-current  generators  can  be  used  to  supply  more  power 
into  the  alternating  circuit  than  is  given  by  their  prime  mover, 
by  receiving  power  from  the  direct-current  side.  In  this  case  a 
part  of  the  alternating  power  is  generated  from  mechanical  power, 
and  the  other  converted  from  direct-current  power,  and  the 
machine  combines  the  features  of  an  alternator  with  those  of  an 
inverted  converter.  Conversely,  when  supplying  direct-current 
power  and  receiving  mechanical  power  from  the  prime  mover  and 
electric  power  from  the  alternating  system,  the  double-current 
generator  combines  the  features  of  a  direct-current  generator  and 
a  converter.  In  either  case  the  armature  reaction,  etc.,  are 
the  sum  of  those  corresponding  to  the  two  types  of  machines 
combined. 

104.  A  combination  of  the  converter  with  the  direct-current 
generator  is  represented  by  the  so-called  "motor  converter,"  which 
consists  of  the  concatenation  of  a  commutating  machine  with  an 
induction  machine. 

If  the  secondary  of  an  induction  machine  is  connected  to  a 
second  induction  or  synchronous  machine  on  the  same  shaft,  and 
of  the  same  number  of  poles,  the  combination  runs  at  half 
synchronous  speed,  and  the  first  induction  machine  as  frequency 
converter  supplies  half  of  its  power  as  electric  power  of  half 
frequency  to  the  second  machine,  and  changes  the  other  half 


262     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

as  motor  into  mechanical  power,  driving  the  second  machine  as 
generator.  (Or,  if  the  two  machines  have  different  number  of 
poles,  or  are  connected  to  run  at  different  speeds,  the  division  of 
power  is  at  a  different  but  constant  ratio) .  Using  thus  a  double- 
current  generator  as  second  machine,  it  receives  half  of  its 
power  mechanically,  by  the  induction  machine  as  motor,  and  the 
other  half  electrically,  by  the  induction  machine  as  frequency 
converter.  Such  a  machine,  then,  is  intermediate  between  a 
converter  and  a  direct-current  generator,  having  an  armature 
reaction  equal  to  half  that  of  a  direct-current  generator. 

Such  motor  converters  have  been  recommended  for  high-fre- 
quency systems,  as  their  commutating  component  is  of  half 
frequency,  and  thus  affords  a  better  commutator  design  than  a 
high-frequency  converter.  They  are  necessarily  much  larger 
than  standard  converters,  but  are  smaller  than  motor  generator 
sets,  as  half  the  power  is  converted  in  either  machine.  One 
advantage  of  this  type  of  machine  for  phase  control  is  that  it 
requires  no  additional  reactive  coils,  as  the  induction  machine 
affords  sufficient  reactance. 

The  use  of  the  converter  to  change  from  alternating  to  alter- 
nating of  a  different  phase,  as,  for  instance,  when  using  a  quarter- 
phase  converter  to  receive  power  by  one  pair  of  its  collector 
rings  from  a  single-phase  circuit  and  supplying  from  its  other 
pair  of  collector  rings  the  other  phase  of  a  quarter-phase  system, 
or  a  three-phase  converter  on  a  single-phase  system  supplying 
the  third  wire  of  a  three-phase  system  from  its  third  collector 
ring, .  is  outside  the  scope  of  this  treatise,  and  is,  moreover,  of 
very  little  importance^  since  induction  or  synchronous  motors 
are  superior  in  this  respect. 

APPENDIX 

Xin.  Direct-current  Converter 

105.  If  n  equidistant  pairs  of  diametrically  opposite  points  of 
a  commutating  machine  armature  are  connected  to  the  ends  of 
n  compensators  or  autotransformers,  that  is,  electric  circuits 
interlinked  with  a  magnetic  circuit,  and  the  centers  of  these  auto- 
transformers  connected  with  each  other  to  a  neutral  point  as  shown 
diagrammatically  in  Fig.  140  for  n  =  3,  this  neutral  is  equidis- 
tant in  potential  from  the  two  sets  of  commutator  brushes,  and 
such  a  machine  can  be  used  as  continuous  current  converter,  to 


SYNCHRONOUS  CONVERTERS 


263 


transform  in  the  ratio  of  potentials  1  :2  or  2  : 1  or  1  : 1,  in  the 
latter  case  transforming  power  from  one  side  of  a  three- wire 
system  to  the  other  side. 

Obviously  either  the  n  autotransformers  can  be  stationary  and 
connected  to  the  armature  by  2  n  collector  rings,  or  the  auto- 
transformers  rotated  with  the  armature  and  their  common  neutral 
connected  to  the  external  circuit  by  one  collector  ring. 

The  distribution  of  potential  and  of  current  in  such  a  direct- 
current  converter  is  shown  in  Fig.  141  for  n  =  2,  that  is,  two 
autotransformers  in  quadrature. 

With  the  voltage  2  e  between  the  outside  conductors  of  the 


FIG.  140. — Diagram  of  direct-current  converter. 

system,  the  voltage  between  the  neutral  and  outside  conductor 
is  ±  e,  that  on  each  of  the  2  n  autotransformer  sections  is 

e  sin(0  —  00  —  — J,  k  =  0,  1,  2  .    .    .  2  n  —  1. 

Neglecting  losses  in  the  converter  and  the  autotransformer,  the 
currents  in  the  two  sets  of  commutator  brushes  are  equal  and  of 
the  same  direction,  that  is,  both  outgoing  or  both  incoming,  and 
opposite  to  the  current  in  the  neutral ;  that  is,  two  equal  currents  i 
enter  the  commutator  brushes  and  issue  as  current  2  i  from  the 
neutral,  or  inversely. 

From  the  law  of  conservation  of  energy  it  follows  that  the  cur- 
rent 2  i  entering  from  the  neutral  divides  in  2  n  equal  and  constant 

branches  of  direct  current,  — ,  in  the  2  n  autotransformer  sections, 

'  n1 

and  hence  enters  the  armature,  to  issue  as  current  i  from  each  of 
the  commutator  brushes. 


264     ELEMENTS  OF  ELECTRICAL  ENGINEERING 
In  reality  the  current  in  each  autotransformer  section  is 

*7*  /  irJf  \ 

--  h  io  \/2  cos  (  e  —  60  ----  h  «)  t 

Ti  \  Ti  I 

where  iQ  is  the  exciting  current  of  the  magnetic  circuit  of  the  auto- 
transformer,  and  a  the  angle  of  hysteretic  advance  of  phase. 
At  the  commutator  the  current  on  the  motor  side  is  larger  than 
the  current  on  the  generator  side,  by  the  amount  required  to 
cover  the  losses  of  power  in  converter  and  autotransformer. 

In  Fig.  141  the  positive  side  of  the  system  is  generator,  the 
negative  side  motor.  This  machine  can  be  considered  as  receiv- 
ing the  current  i  at  the  voltage  e  from  the  negative  side  of  the 
system,  and  transforming  it  into  current  i  at  voltage  e  on  the 


21 


FIG.  141. — Distribution  e.m.f .  and  current  in  direct-current  converter. 

positive  side  of  the  system,  or  it  can  be  considered  as  receiving 
current  i  at  voltage  2e  from  the  system,  and  transforming  it 
into  current  2  i  at  the  voltage  e  on  the  positive  side  of  the  system, 
or  of  receiving  current  2  i  at  voltage  e  from  the  negative  side,  and 
returning  current  i  at  voltage  2e.  In  either  case  the  direct- 
current  converter  produces  a  difference  of  power  of  2  ie  between 
the  two  sides  of  the  three- wire  system. 

The  armature  reaction  of  the  currents  from  the  generator  side 
of  the  converter  is  equal  but  opposite  to  the  armature  reaction 
of  the  corresponding  currents  entering  the  motor  side,  and  the 
motor  and  generator  armature  reactions  thus  neutralize  each 
other,  as  in  the  synchronous  converter;  that  is,  the  resultant 


SYNCHRONOUS  CONVERTERS 


265 


armature  reaction  of  the  continuous-current  converter  is  prac- 
tically zero,  or  the  only  remaining  armature  reaction  is  that 
corresponding  to  the  relatively  small  current  required  to  rotate 
the  machine,  that  is,  to  supply  the  internal  losses  in  the  same. 
The  armature  reaction  of  the  current  supplying  the  electric 
power  transformed  into  mechanical  power  obviously  also  remains, 
if  the  machine  is  used  simultaneously  as  motor,  as  for  driving  a 
booster  connected  into  the  system  to  produce  a  difference  between 
the  voltages  of  the  two  sides,  or  the  armature  reaction  of  the 
currents  generated  from  mechanical  power  if  the  machine  is 
driven  as  generator. 


I  I.  Ill 


jiHojifljybJU  yyyyyyyyyyyyyyywyy  jyyyyy 

a'"o"o  a^aza3 

C 


B 


J1 


a"a"a 


1J. 


2i 


FIG.  142. — Development  of  a  direct-current  converter. 

106.  While  the  currents  in  the  armature  coils  are  more  or  less 
sine  waves  in  the  alternator,  rectangular  reversed  currents  in 
the  direct-current  generator  or  motor,  and  distorted  triple-fre- 
quency currents  in  the  synchronous  converter,  the  currents  in  the 
armature  coils  of  the  direct-current  converter  are  approximately 
triangular  double-frequency  waves. 

Let  Fig.  142  represent  a  development  of  a  direct-current  con- 
verter with  brushes  BI  and  B2,  and  C  one  autotransformer  re- 
ceiving current  2  i  from  the  neutral.  Consider  first  an  armature 
coil  ai  adjacent  and  behind  (in  the  direction  of  rotation)  an  auto- 
transformer  lead  61.  In  the  moment  when  autotransformer  leads 
61  62  coincide  with  the  brushes  BI  B%  the  current  i  directly  enters 


266     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

the  brushes  and  coil  ai  is  without  current.  In  the  next  moment 
(Fig.  142A)  the  total  current  i  from  61  passes  coil  ai  to  brush  Bit 
while  there  is  yet  practically  no  current  from  61  over  coils  a' 
a",  etc.,  to  brush  Bz.  But  with  the  forward  motion  of  the  arma- 
ture less  and  less  of  the  current  from  61  passes  through  «i  a2, 
etc.,  to  brush  BI  and  more  over  a'  a",  etc.,  to  brush  B^  until  in 
the  position  of  a\  midway  between  61  and  62  (Fig.  1425),  one-half 
of  the  current  from  61  passes  ai  a2,  etc.,  to  BI,  the  other  half  a' 
a",  etc.,  to  B%.  With  the  further  rotation  the  current  in  a\ 
grows  less  and  becomes  zero  when  61  coincides  with  B%,  or  half 
a  cycle  after  its  coincidence  with  BI.  That  is,  the  current  in 


FIG.  143. — Current  in  the  various  coils  of  a  direct-current  converter. 

coil  ai  approximately  has  the  triangular  form  shown  as  ii  in  Fig. 
143,  changing  twice  per  period  from  0  to  i.  It  is  shown  negative, 
since  it  is  against  the  direction  of  rotation  of  the  armature.  In 
the  same  way  we  see  that  the  current  in  the  coil  a',  adjacent 
ahead  of  the  lead  61,  has  a  shape  shown  as  i'  in  Fig.  143.  The 
current  in  coil  a0  midway  between  two  commutator  leads  has  the 
form  io,  and  in  general  the  current  in  any  armature  coil  ax,  dis- 
tant by  angle  r  from  the  midway  position  a0,  has  the  form 
ix,  Fig.  143. 

All  the  currents  become  zero  at  the  moment  when  the  autotrans- 
former  leads  61  62  coincide  with  the  brushes  BI  B2,  and  change 


SYNCHRONOUS  CONVERTERS 


267 


by  i  at  the  moment  when  their  respective  coils  pass  a  commu- 
tator brush.  Thus  the  lines  A  and  A'  in  Fig.  144  with  zero 
values  at  BI  B^  the  position  of  brushes,  represent  the  currents  in 
the  individual  armature  coils.  The  current  changes  from  A  to 
A'  at  the  moment  0  =  r  when  the  respective  armature  coil 
passes  the  brush,  twice  per  period.  Due  to  the  inductance  of 
the  armature  coils,  which  opposes  the  change  of  current,  the 
current  waves  are  not  perfectly  triangular,  but  differ  somewhat 
therefrom. 

With  n  autotransformers,  each  autotransformer  lead  carries  the 

current  — ,  which  passes  through  the  armature  coils  as  triangular 

n 

current,  changing  by  —  in  the  moment  the  armature  coil  passes 

a  commutator  brush.  This  current  passes  the  zero  value  in  the 
moment  the  autotransformer  lead  coincides  with  a  brush.  Thus, 


FIG.  144. — Current  in  individual  coils  of  a  direct-current  converter  with  one 

compensator. 

the  differents  current  of  n  autotransformers  which  are  superposed 
in  an  armature  coil  ax  have  the  shape  shown  in  Fig.  199  for  n  =  3. 
That  is,  each  autotransformer  gives  a  set  of  slanting  lines 
AiA'i,  A^A'z,  AsA's,  and  all  the  branch  currents  i\,  iz,  iz,  super- 
posed, give  a  resultant  current  ix,  which  changes  by  i  in  the 
moment  the  coil  passes  the  brush.  ix  varies  between  the  extreme 

values  £  (2  p  —  1)  and  ^(2p  +  1),  if  the  armature  coil  is  dis- 
placed from  the  midway  position  between  two  adjacent  autotrans- 
former leads  by  angle  r,  and  p  =  -  •  p  varies  between  —  ^— 


and 


Thus  the  current  in  an  armature  coil  in  position  p  =  -  can 


268     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

be  denoted  in  the  range  from  p  to  1  +  p,  or  r  to  TT  +  r,  by 


where 


e 

x  =  - 

7T 


\\  /  "1?t  A'f  A",  /      A"2 


A 


FIG.  145. — Current  in  a  single  coil  of  a  direct-current  converter  with  three 

compensators. 


The  effective  value  of  this  current  is 
/  = 


Since  in  the  same  machine   as   direct-current   generator  at 
voltage  2  e  and  current  i]  the  current  per  armature  coil  is  ~>  the 

ratio  of  current  is 

I 
i 
2 


SYNCHRONOUS  CONVERTERS 


269 


and  thus  the  relative  Pr  loss  or  the  heat  developed  in  the  armature 
coil, 

/A2 


*-  i 


with  a  minimum, 
and  a  maximum, 


p  =  0,  TO  =  M> 

1 


^ 1 =  — ' 

Jm       3^w2         3w2 

The  mean  heating  or  Pr  of  the  armature  is  found  by  integrating 
over  7  from 

1  1 


as 


r  =  n 


J_ 

2n 


ydp 


2n 

I_L  JL^  _  1  +  n2 
3n2  ' 


This  gives  the  following  table,  for  the  direct-current  converter, 
of  minimum  current  heating,  70,  in  the  coil  midway  between 


DIRECT-CURRENT  CONVERTER  IV  RATING 

d.  c. 

No.  of  compensators,  n  = 

gen. 

1 

2 

3 

4 

n 

00 

Minimum  current  heating 

P  =  0,                        70  = 

1  ' 

K 

K 

H 

H 

H 

H 

Maximum  current  heating, 

1 

i 

1 

% 

J^2 

^ 

i^g 

/^ 

n' 

n 

Mean  current  heating, 

1 

r  = 

1 

H 

^2 

!%7 

1%8 

H 

Rating,                       1 

Vr" 

1 

1.225 

1.549 

1.643 

1.681 

/   3n2 

1.732 

\l+n2 

270     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

adjacent  commutator  leads,  maximum  current   heating,  ym,  in 
the  coil  adjacent  to  the  commutator  lead,  mean  current  heating, 
r,  and  rating  as  based  on  mean  current  heating  in  the  armature, 
1   . 


As  seen,  the  output  of  the  direct-current  converter  is  greater 
than  that  of  the  same  machine  as  generator.  Using  more  than 
three  autotransformers  offers  very  little  advantage,  and  the  dif- 
ference between  three  and  two  autotransformers  is  comparatively 
small,  also,  but  the  difference  between  two  and  one  autotransfor- 
mer,  especially  regarding  the  local  armature  heating,  is  considera- 
ble, so  that  for  most  practical  purposes  a  two-autotransformer 
converter  would  be  preferable. 

The  number  of  autotransformers  used  in  the  direct-current 
converter  has  a  similar  effect  regarding  current  distribution, 
heating,  etc.,  as  the  number  of  phases  in  the  synchronous 
converter. 

Obviously  these  relative  outputs  given  in  above  table  refer  to 
the  armature  heating  only.  Regarding  commutation,  the  total 
current  at  the  brushes  is  the  same  in  the  converter  as  in  the 
generator,  the  only  advantage  of  the  former  being  the  better 
commutation  due  to  the  absence  of  armature  reaction. 

The  limit  of  output  set  by  armature  reaction  and  correspond- 
ing field  excitation  in  a  motor  or  generator  obviously  does  not 
exist  at  all  in  a  converter.  It  follows  herefrom  that  a  direct- 
current  motor  or  generator  does  not  give  the  most  advantageous 
direct-current  converter,  but  that  in  the  direct-current  converter 
just  as  in  the  synchronous  converter,  it  is  preferable  to  propor- 
tion the  parts  differently  in  accordance  with  above  discussion, 
as,  for  instance,  to  use  less  conductor  section,  a  greater  number  of 
conductors  in  series  per  pole,  etc. 

XIV.  Three-wire   Generator  and  Converter 

107.  A  machine  based  upon  the  principle  of  the  direct-current 
converter  is  frequently  used  to  supply  a  three-  wire  direct-current 
distribution  system  (Edison  system).  This  machine  may  be  a 
single  generator  or  synchronous  converter,  which  is  designed  for 
the  voltage  between  the  outside  conductors  of  the  circuit  (the 
positive  and  the  negative  conductor),  220  to  280  volts,  while  the 
middle  conductor  of  the  system,  or  neutral  conductor,  is  con- 


SYNCHRONOUS  CONVERTERS 


271 


nected  to  the  generator  by  autotransformer  and  collector  rings, 
or,  in  the  case  of  a  synchronous  converter,  is  connected  to  the 
neutral  of  the  step-up  transformers,  and  the  latter  thus  used 
as  autotransformers. 


-*; 

to     

v        2  "* 

n 

C, 

2                                0         1        , 

o 

)T               ' 

k    g 

to       t.<  

2 

FIG.  146. — Three-wire  machine  with  single  autotransformer. 

A  three-wire  generator  thus  is  a  combination  of  a  direct- 
current  generator  and  a  direct-current  converter,  and  a  three- 
wire  converter  is  a  combination  of  a  synchronous  converter  and 
a  direct-current  converter.  Such  a  three-wire  machine  has  the 
advantage  over  two  separate  machines,  connected  to  the  two 


2C 


i-l.   < 

Fro.  147. — Three-wire  system  with  two  machines. 

sides  of  the  three- wire  direct-current  system,  of  combining  two 
smaller  machines  into  one  of  twice  the  size,  and  thus  higher 
space-  and  operation-economy  and  lower  cost,  and  has  the  further 
advantage  that  only  half  as  large  current  is  commutated  as  by 


272     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

the  use  of  two  separate  machines;  that  is,  the  positive  brush 
of  the  machine  on  the 'negative  and  the  negative  brush  of  the 
machine  on  the  positive  side  of  the  system  are  saved,  as  seen  by 
the  diagrammatic  sketch  of  the  machine  in  Fig.  146  and  the  two 
separate  two-wire  machines  in  Fig.  147.  The  use  of  three-wire 
220-volt  machines  on  three-wire  direct-current  systems  thus  has 
practically  displaced  that  of  two  separate  110- volt  machines. 

A.  THREE-WIRE  DIRECT-CURRENT  GENERATOR 

108.  In  such  machines,  either  only  one  compensator  or  auto- 
transformer  is  used  for  deriving  the  neutral,  as  shown  diagram- 
matically  in  Fig.  146,  or  two  autotransformers  in  quadrature,  as 
shown  in  Fig.  148,  but  rarely  more. 


FIG.  148. — Three-wire  machine  with  two  compensators. 

As  the  efficiency  of  conversion  of  a  direct-current  converter 
with  two  autotransformers  in  quadrature  (Fig.  148)  is  higher  than 
that  of  a  direct-current  converter  with  single  autotransformer 
(Fig.  146),  it  is  preferable  to  use  two  (or  even  more)  autotrans- 
formers where  a  large  amount  of  power  is  to  be  converted,  that 
is,  where  a  very  great  unbalancing  between  the  two  sides  of  the 
three-wire  system  may  occur,  or  one  side  may  be  practically 
unloaded  while  the  other  is  overloaded.  Where,  however,  the 
load  is  fairly  distributed  between  the  two  sides  of  the  system, 
that  is,  the  neutral  current  (which  is  the  difference  between  the 
currents  on  the  two  sides  of  the  system)  is  small  and  so  only  a 
small  part  of  the  generator  power  is  converted  from  one  side  to 
the  other,  and  the  efficiency  of  this  conversion  thus  of  negligible 


SYNCHRONOUS  CONVERTERS        273 

influence  on  the  heating  and  the  output  of  the  machine,  a  single 
autotransformer  is  preferable  because  of  its  simplicity.  In 
three-  wire  distribution  systems  the  latter  is  practically  always  the 
case,  that  is,  the  load  fairly  balanced  and  the  neutral  current 
small. 

The  size  of  the  autotransformers  depends  upon  the  amount  of 
unbalanced  power,  that  is,  the  maximum  difference  between  the 
load  on  the  two  sides  of  the  three-  wire  system,  and  thus  equals 
the  product  of  neutral  current  iQ  and  voltage  e  between  neutral 
and  outside  conductor;  that  is,  in  the  three-  wire  system  of  vol- 
tage e  per  circuit,  voltage  2e  between  the  outside  conductors, 
and  maximum  current  i  in  the  outside  conductors,  the  generator 
power  rating  is 

p  =  2  ei. 

Let  now  io  =  maximum  unbalanced  current  in  the  neutral  — 
usually  not  exceeding  10  to  20  per  cent,  of  i  —  and  using  a  single 
autotransformer,  connected  diametrically  across  the  armature, 
Fig.  146,  the  maximum  of  the  alternating  voltage  which  it  re- 
ceives is  2  e,  and  its  effective  voltage  therefore  e  \/2.  As  the 
neutral  current  iQ  divides  when  entering  the  autotransformer,  the 

current  in  the  compensating  winding  is  -^  (neglecting  the  small 

z 

exciting  current),  and  the  volt-ampere  capacity  of  the  autotrans- 
former thus  is 


and 

PQ  _        1       io 

P   ~  2  V2  i 

=  0.354  *°- 

x 

Even  with  the  neutral  current  equal  to  the  current  in  the  out- 
side conductor,  or  the  one  side  of  the  system  fully  loaded,  the 
other  not  loaded,  the  autotransformer  thus  would  have  only 
35.4  per  cent,  of  the  volt-ampere  capacity  of  the  generator,  and 
as  an  autotransformer  of  ratio  1  -r-  1  is  half  the  size  of  a  trans- 
former of  the  same  volt-ampere  capacity,  in  this  case  the  auto- 
transformer has,  approximately,  the  size  of  a  transformer  of 
17.7  per  cent,  of  the  size  of  the  generator. 

With  the  maximum  unbalancing  of  20  per  cent.,  or  -r-  =  0.2, 


274     ELEMENTS  OF  ELECTRICAL    ENGINEERING 

the  autotransformer  thus  has  7  per  cent,  of  the  volt-ampere 
capacity  of  the  generator,  or  the  size  of  a  transformer  of  only 
3.5  per  cent,  of  the  generator  capacity,  that  is,  is  very  small,  and 
this  method  is  therefore  the  most  convenient  for  deriving  the 
neutral  of  a  three- wire  distribution  system. 

When  using  n  autotransformers,  obviously  each  has  -  of  the 

size  which  a  single  autotransformer  would  have. 

The  disadvantage  of  the  three-wire  generator  over  two  sepa- 
rate generators  is  that  a  three-wire  generator  can  only  divide 
the  voltage  in  two  equal  parts,  that  is,  the  two  sides  of  the  system 
have  the  same  voltage  at  the  generator.  The  use  of  two  separate 
generators,  however,  permits  the  production  of  a  higher  voltage 
on  one  side  of  the  system  than  on  the  other,  and  thus  takes  care 
of  the  greater  line  drop  on  the  more  evenly  loaded  side.  Even 
in  the  case,  however,  where  a  voltage  difference  between  the  two 
sides  of  the  system  is  desired  for  controlling  feeder  drops,  it  can 
more  economically  be  given  by  a  separate  booster  in  the  neu- 
tral, as  such  a  booster  woul'd  require  only  a  capacity  equal  to 
the  neutral  current  times  half  the  desired  voltage  difference 
between  the  two  sides,  and  with  20  per  cent,  neutral  current  and 
10  per  cent,  voltage  difference  between  the  two  sides,  thus  would 
have  only  1  per  cent,  of  the  size  of  the  generator. 

B.  THBEE-WIRE  CONVERTER 

109.  In  a  converter  feeding  a  three-wire  direct-current  system 
the  neutral  can  be  derived  by  connection  to  the  transformer 
neutral.  Even  in  this  case,  however,  frequently  a  separate  auto- 
transformer is  used,  connected  across  a  pair  of  collector  rings  of 
the  converter,  since,  as  seen  above,  with  the  moderate  unbalanc- 
ing usually  existing,  such  a  compensator  is  very  small. 

When  connecting  the  direct-current  neutral  to  the  transformer 
neutral  it  is  necessary  to  use  such  a  connection  that  the  trans- 
former can  operate  as  autotransformer,  that  is,  that  the  direct 
current  in  each  transformer  divides  into  two  branches  of  equal 
m.m.f.,  otherwise  the  direct-current  produces  a  unidirectional 
magnetization  in  the  transformer,  which  superimposed  upon 
the  magnetic  cycle  raises  the  magnetic  induction  beyond  satura- 
tion, and  thus  causes  excessive  exciting  current  and  heating, 
except  when  very  small. 


SYNCHRONOUS  CONVERTERS 


275 


k       *T.     \ 


FIG.  149. — Neutral  of  Y-connected  transformers  connected  to  neutral  of 
three-wire  system  supplied  from  a  three-phase  converter. 


FIG.  150. — Quarter-phase  converter  with  transformer  neutral  connected  to 
direct-current  neutral. 


FIG.  151. — Three-phase  converter  with  neutral  of  the  T-connected  trans- 
formers as  direct-current  neutral. 


FIG.  152. — Three-phase  converter  with  transformer  neutral  connected  to 
direct-current  neutral. 


276     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

For  instance,  with  Y  connection  of  the  transformers  supplying 
a  three-phase  converter,  Fig.  149,  each  transformer  secondary 
receives  one-third  of  the  neutral  current,  and  if  this  current  is  not 
very  small  and  comparable  with  the  exciting  current  of  the 
transformer — which  can  rarely  be — the  magnetic  density  in 
the  transformer  rises  beyond  saturation  by  this  unidirectional 
m.m.f.  This  connection  thus  is  in  general  not  permissible  for 
deriving  the  neutral. 

In  a  quarter-phase  converter,  as  shown  in  Fig.  150,  the  trans- 
former neutral  can  be  used  as  direct-current  neutral,  since 
in  each  transformer  the  direct  current  divides  into  two  equal 
branches,  which  magnetize  in  opposite  direction,  and  so  neu- 
tralize. 

The  T  connection,  Fig.  151,  can  be  used  for  three-phase  con- 
verters with  the  neutral  derived  from  a  point  at  one-third  the 
height  of  the  teaser  transformer,  as  seen  in  Fig.  151. 

Delta  connection  on  three-phase  and  double  delta  on  six-phase 
converters  cannot  be  used,  as  it  has  no  neutral,  but  in  this  case  a 
separate  compensator  is  required. 

The  diagrammatical  connections  of  transformers  can,  however, 
be  used  on  six-phase  converters,  and  the  connection  shown  in 
Fig.  152,  which  has  two  coils  on  each  transformer,  connected  to 
different  phases,  on  three-phase  converters. 


D.  ALTERNATING-CURRENT  TRANSFORMER 
I.  General 

110.  The  alternating-current  transformer  consists  of  a  magnetic 
circuit  interlinked  with  two  electric  circuits,  the  primary,  which 
receives  power,  and  the  secondary,  which  gives  out  power. 

Since  the  same  magnetic  flux  interlinks  primary  and  second- 
ary turns,  the  same  voltage  is  induced  in  every  turn  of  the 
electric  circuits,  and  the  e.m.fs.  induced  in  the  primary  and 
in  the  secondary  winding  therefore  have  the  ratio  of  turns: 

«'i       ni 
—r  —  —  =  a. 
.      e'2       n2 

This  ratio  is  called  the  ratio  of  transformation. 

The  ratio  of  transformation  of  a  transformer  is  the  ratio  of  turns 
of  primary  and  secondary  windings. 

In  addition  to  the  induced  e.m.fs.  e'i  and  e\,  resistance  r  and 
reactance  x  consume  voltage  in  primary  and  secondary  wind- 
ings. The  voltage  consumed  by  the  resistance  represents  waste  of 
power;  the  voltage  consumed  by  reactance  is  wattless,  but  causes 
lag  of  current,  that  is,  lowers  the  power  factor;  while  the  in- 
duced voltages  give  the  power  transfer  from  primary  to  sec- 
ondary. Efficiency  therefore  requires  to  make  the  former  vol- 
tages as  small  as  possible,  and  the  induced  voltages  as  near  to 
the  terminal  voltages  as  possible.  Therefore,  in  first  approxi- 
mation, the  ratio  of  the  terminal  voltages  e\  and  e%  is  the  ratio 
of  transformation: 


As,  approximately,  the  power  output  of  the  secondary  equals  the 
power  input  into  the  primary,  it  is: 


hence, 

ti       1 


277 


278     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

that  is,  the  transformer  changes  from  voltage  e\  and  current 
i\  to  voltage  62  =  —  and  current  iz  =  aii. 

In  general  either  of  the  two  transformer  circuits  may  be  used 
as  primary  or  as  secondary,  and  by  their  use  transformers  thus 
are  distinguished  as  step-down  transformers,  if  the  primary 
voltage  is  higher  than  the  secondary,  and  step-up  transformers, 
if  the  secondary  voltage  is  higher.  Instead  of  the  expression 
"primary"  and  "secondary,"  constructively  it  therefore  is 
preferable  to  speak  of  "high  voltage  winding"  and  "low  voltage 
winding." 

111.  The  foremost  use  of  the  transformer  therefore  is  for 
changing  of  the  voltage: 

From  the  medium  high  primary  distribution  voltage  (2300) 
to  the  low  secondary  consumer  voltage  (110,  220). 

From  the  high  transmission  (30  to  150  kilovolts)  to  the  primary 
distribution  voltage  (2300)  or  the  voltage  required  by  syn- 
chronous motor,  synchronous  converter,  etc. 

From  the  low  or  medium  high  generator  voltage  to  the  high 
transmission  voltage. 

Other  occasional  uses  of  transformers  are: 

To  electrically  tie  systems  together,  so  as  to  permit  exchange 
of  power  between  them,  and  synchronous  operation.  In  this 
case,  depending  on  the  distribution  of  the  load  in  the  system, 
either  transformer  winding  may  be  primary  or  secondary. 

To  break  up  electrically  a  very  large  system,  so  that  a  ground 
in  one  part  does  not  ground  the  entire  system.  In  this  case, 
the  transformer  ratio  usually  is  1  -f-  1. 

In  all  these  cases,  the  transformers  are  "constant  potential 
transformers,"  that  is,  primary  and  thus  secondary  voltage  are 
constant  or  approximately  so. 

Transformers  supplied  with  constant  current  in  the  primary 
give  practically  constant  current  in  the  secondary,  at  a  primary 
voltage  varying  with  the  secondary  voltage.  Such  transformers 
are  used  in  constant-current  circuits,  for  supplying  meters  in 
high  voltage  circuits,  etc. 

Further  uses  of  transformers  are  for  operating  instruments, 
switches,  etc.,  in  high  voltage  systems.  In  this  case,  the  trans- 
formers may  be  potential  transformers — connected  across  the 
constant  voltage  circuit,  or  current  transformers — connected 
in  series  into  the  circuit,  for  the  supply  of  meters,  the  opera- 
tion of  overload  circuit  breakers,  etc. 


ALTERNATING-CURRENT  TRANSFORMER       279 

• 

Where  not  expressly  stated  otherwise,  in  general  a  constant 
potential  transformer  is  understood. 

II.  Excitation 

112.  The    primary    current    i\    is    not    strictly    proportional 
to  the  secondary  current,  i2  by  the  ratio  of  transformation, 


TRANSFORMER 
Excitation  and  Iron  Losses 


Vo 


tage 


fower 
factor 


50 


30 


FIG.  153. — Excitation  and  core  loss  of  transformer. 

and  does  not  become  zero  at  no  load  or  open  circuit,  but  a  small 
and  lagging  current  ^o  remains  at  no  load,  which  is  called  the 
exciting  current.  It  produces  the  magnetic  flux  and  supplies 
the  losses  in  the  iron,  so-called  "core  loss."  Its  reactive  com- 
ponent, imj  is  called  the  magnetizing  current,  and  is  usually 
greatly  distorted  in  wave  shape,  while  the  energy  component, 


280     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

ih,  does  not  much  differ  from  a  sine  wave,  and  is  the  hysteresis 
energy  current: 

/o  =  ih  -  Jim- 

Under  load,  the  primary  current  then  consists  of  two  com- 
ponents: the  load  current  7'2  which  is  the  transformed  second- 

ary current  7'2  =  —  >  and  the  exciting,  current  IQ.     The  total 
«i 

primary  current  thus  is: 

Ji  =  /'2  +  /o  =  ^+  (ih-jim). 

In  general,  /0  rarely  exceeds  5  per  cent,  of  the  full-load  primary 
current. 

Core  loss  and  exciting  current,  with  its  two  components,  are 
determined  by  measuring  volts,  amperes  and  watts  input  into 
the  primary  of  the  transformer  at  open  secondary.  It  is  ob- 
vious that  either  of  the  transformer  coils  can  for  this  purpose  be 
used  as  primary,  and  usually  the  low  voltage  coil  is  employed 
as  more  convenient. 

Such  excitation  and  core-loss  curves  are  given  in  Fig.  153, 
with  the  impressed  volts  as  abscissae,  and  the  total  exciting 
current,  and  core  loss  as  ordinates. 

The  exciting  current  is  usually  not  proportional  to  the  voltage, 
due  to  the  use  of  a  closed  magnetic  circuit,  and  for  the  same 
reason,  the  power-factor  of  the  exciting  current  is  fairly  high, 
from  40  to  60  per  cent.,  except  at  high  voltages,  where  magnetic 
saturation  causes  an  abnormal  increase  of  the  magnetizing 
current. 

The  power-factor  is  shown  on  Fig.  153. 

IE.  Losses  and  Efficiency 

113.  The  losses  in  the  transformer  are 

(a)  The  core  loss,  comprising  the  loss  by  hysteresis  and 
eddy  currents  in  the  iron.  This  depends  on  the  maximum 
magnetic  flux,  and  thus  on  the  induced  voltage: 


and  as  the  induced  voltage  is  practically  equal  to  the  impressed 
voltage  61,  at  constant  impressed  voltage,  the  core  loss  is  practi- 
cally constant,  and  is  often  assumed  as  constant,  that  is,  the 


ALTERNATING-CURRENT  TRANSFORMER       281 

core  loss  is  a  constant  or  no-load  loss,  and  is  supplied  by  the 
exciting  current  i0. 

(b)  The  i2r  losses  in  the  primary  and  secondary  coils.     These 
are  load  losses,  increasing  with  the  square  of  the  load. 

(c)  Spurious  load  losses,  as  eddy  currents  in  the  conductors 
and  other  metal  parts.     With  proper   design  these  should  be 
negligible. 

(d)  In   very   high   voltage   transformers,   electrostatic   losses 
in  the  insulation  appear.     These  usually  are  small  in  large  well- 
designed  transformers. 

In  large  transformers,  the  total  &r  loss  may  be  less  than  1 
per  cent.,  and  so  also  the  core  loss,  resulting  in  efficiencies  of 
over  98  per  cent. 

As  instance  are  shown,  in  Figs.  154  and  155,  the  loss  curves 
and  the  efficiency  curves  of  two  transformers,  of  the  respective 
constants,  at  full  load  of  20  kw. 


I.  Low  core-loss  type, 
Fig.  154 

II.  Low  t*r  loss  type, 
Fig.  155 

Exciting  current  

4  per  cent. 

4      per  cent. 

Primary  resistance  loss  

1  per  cent. 

0  .  5  per  cent. 

Secondary  resistance  loss  
Core  loss  

1  per  cent. 
1  per  cent. 

0  .  5  per  cent. 
2      per  cent. 

For  convenience,  exciting  current  and  losses  are  frequently 
given  in  per  cent,  of  the  full-load  output  of  the  transformer. 

The  curves  correspond  to  non-inductive  load.  The  core  loss 
comprises  hysteresis,  which  varies  with  the  1.6  power  of  the 
induced  voltage  and  eddies  proportional  to  the  square  of  induced 
voltage.  Hence,  within  the  narrow  range  of  variation  of  the 
induced  voltage  between  no  load  and  full  load  of  a  constant  poten- 
tial transformer,  the  core  loss  can  be  approximated  as  propor- 
tional to  the  1.7  power  of  the  induced  voltage.  The  induced 
voltage  at  non-inductive  load  equals  impressed  voltage  minus 
primary  ir,  when  neglecting  the  inductive  drop,  which  is  permis- 
sible at  non-inductive  load.  As  the  induced  voltage  thus  de- 
creases proportional  to  primary  ir,  the  core  loss  decreases  pro- 
portional to  1.7  times  the  primary  ir.  Thus,  with  the  primary 
tV  equal  to  1  per  cent,  at  full  load,  the  induced  voltage  has 


282     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

decreased  1  per  cent,  and  the  core  loss  1.7  per  cent,  at  full  load, 
and  correspondingly  at  other  loads. 

As  seen,  I  and  II  have  the  same  full-load  efficiency,  but  II  is 
more  efficient  at  overload,  I  at  partial  load. 


EFFICIENCY  and    LOSSES 
of 

Low  Corcloss  Transformer 
1%   Iron  Loss 
2%  i2r  Loss 


.9       1.0      1.1      1.2      1.3      1.4      1.5 


.1        .2       .3       .4        .5       .6       .7        .8 

FIG.  154. — Efficiency  and  losses  of  low  core  loss  transformer. 


114.  In  transformers  for  lighting  and  general  distribution 
{usually  with  2300  volt  primary  and  2  X  115  volt  secondary) 
the  transformer  is  generally  heavily  loaded  only  for  a  short  time 
during  the  day,  partly  loaded  for  a  moderate  time,  and  prac- 
tically unloaded  for  most  of  the  time.  Thus  load  curves  of  such 
a  transformer  would  be: 


ALTERNATING-CURRENT  TRANSFORMER       283 


A. 

Lighting  and  power                                                B.  Lighting  only 

2 

hours 

at  IK 

load. 

2 

hours 

at  IK 

load. 

2 

hours 

at    % 

load. 

2 

hours 

at    % 

load. 

6 

hours 

at    Y2 

load. 

20 

hours 

of         ]/ 

clL      /2  G 

load. 

14 

hours 

at    Ho 

load. 

EFFICIENCY  and  LOSSES 

of 
High  Coreloss  Transformer; 
2  %     Iron  Loss 
1%    t2rLoss 

I 

V 

V 

A 

\ 

\ 

% 

\ 

% 

100 

> 

Loss 
5.0 

90 

\ 

,^~- 

\ 

*j 

4.5 

80/ 

/* 

\ 

s. 

/ 

/ 

4.0 

/ 

5 

X. 

^X 

x^ 

3.5 

^> 

^ 

^^ 

^ 

x^ 

3.0 

-^o^ 

*\* 

se»^ 

•^^ 

2^, 

—  —  — 

—  = 

=r 

_    — 

—  - 

^.  —  • 

^  • 

Ire 

nLos 

ses 

S 

/ 

^ 

x 

2.0 

1.5 

^^ 

<; 

1.0 

j2 

V-O^ 

^ 

.5 

u_i— 

Lc 

ad^- 

->- 

.1       .2       .3       .4       .5       .6       .7       .8       .9       1.0      1.1      1.2      1.3     1.4      1.5 

FIG.  155. — Efficiency  and  losses  of  low  ilr  loss  transformer. 


284     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

This  gives  for  the  two  types  of  transformers: 
A.  LIGHTING  AND  POWER 


Time 

Load 

=  Per  cent. 

TimeX 
load 

I 

II 

Losses 

Time  X 
losses 

Losses 

Time  X 
losses 

2hr. 

IH 

125 

250 

4.10 

8.20 

3.54 

7.08 

2hr. 

% 

75 

150 

2.11 

4.22 

2.55 

5.10 

6hr. 

H 

50 

300 

1.50 

9.00 

2.25 

13.50 

14  hr. 

Y20 

5 

S  = 

70 

1.00 

14.00 

2.00 

28.00 

770 

35.42 

53.68 

Input                            805  .  42                    823  .  68 

Per  cent,  loss                  4.41                        6.51 

Per  cent,  efficiency       95  .59                      93  .  49 

B.  LIGHTING  ONLY 


I 

II 

Time 

Load 

=  Percent. 

Time  X 
load 

Losses 

Time  X 
losses 

Losses 

Time  X 
losses 

2hr. 

IK 

125 

250 

4.10 

8.20 

3.54 

7.08 

2hr. 

H 

75 

150 

2.11 

4.22 

2.55 

5.10 

20  hr. 

Mo 

5 

100 

1.00 

20.00 

2.00 

40.00 

S  = 

500 

32.42 

52.18 

Input                            532.42                     552.18 

Per  cent,  loss                   6.11                        9  .  45 

Per  cent,  efficiency       93  .89                       90  .  55 

As  seen,  while  I  and  II  have  the  same  full-load  efficiency, 
97.1  per  cent.,  I,  the  low  core-loss  type,  gives  a  much  higher 
all-day  efficiency,  the  more  so  the  shorter  the  time  of  heavy  load, 
that  is,  is  far  preferable  for  general  distribution,  as  "lighting 
transformer." 

Inversely,  in  large  power  transformers  in  transmission  systems, 
the  high  partial  load  efficiency  of  the  low  core-loss  type  is  of  less 
importance,  as  such  transformers  are  usually  not  run  at  partial 
load,  but  with  a  decrease  of  load  on  the  system,  transformers  and 
generators  are  cut  out  and  the  remaining  ones  kept  loaded.  Of 


ALTERNATING-CURRENT  TRANSFORMER       285 

importance,  however,  is  low  i2r  loss.  Under  emergency  conditions 
requiring  overloading  of  some  transformer,  the  increased  loss 
is  all  in  the  copper,  and  the  less  therefore  the  i2ry  the  less  is 
the  danger  of  destruction  by  overheating  in  a  case  of  a  temporary 
overload.  Thus  the  low  izr  loss  type  of  transformer  is  preferable 
for  large  power  units. 

IV.  Regulation 

115.  As  primary  and  secondary  winding  of  the  transformer  can- 
not occupy  the  same  space,  and  in  addition  some  insulation — 
more  or  less  depending  on  the  voltage — must  be  between  them, 
there  is  thus  a  space  between  primary  and  secondary  through 
which  the  primary  current  can  send  magnetic  flux  which  does 
not  interlink  with  the  secondary  winding,  but  is  a  self-induc- 
tive or  leakage  flux  and  in  the  same  manner  the  secondary 
current  sends  self-inductive  or  leakage  flux  through  the  space 
between  primary  and  secondary  winding.  These  fluxes  give 
rise  to  the  self -inductive  or  leakage  reactances  x\  and  Xz  of  the 
transformer. 

Or  in  other  words,  two  paths  exist  for  magnetic  flux  in  the 
transformer:  the  path  surrounding  primary  and  secondary  coils, 
through  which  flows  the  mutual  magnetic  flux  of  the  transformer, 
which  is  the  useful  flux,  that  is,  the  flux  which  transfers  the  power 
from  primary  to  secondary  circuit;  and  the  space  between  pri- 
mary and  secondary  winding  through  which  the  self-inductive 
or  leakage  flux  passes,  that  is,  the  flux  interlinked  with  one  wind- 
ing only,  but  not  the  other  one.  The  latter  flux  thus  does  not 
transmit  power,  but  consumes  reactive  voltage  and  thereby  pro- 
duces a  voltage  drop  and  a  lag  of  the  current  behind  the  voltage, 
that  is,  is  in  general  objectionable. 

The  mutual  magnetic  flux  passes  through  a  closed  magnetic 
circuit,  with  the  (vector)  difference  between  primary  and  second- 
ary current,  that  is,  the  exciting  current  J0  =  /i as  m.m.f. 

The  self-inductive  flux  passes  through  an  open  magnetic  circuit 
of  high  reluctance,  the  narrow  space  between  primary  and 
secondary  windings,  but  it  is  due  to  the  full  m.m.f.  of  primary 
or  secondary  current  and,  therefore,  in  spite  of  the  high  reluctance 
of  the  leakage  flux  path  due  to  the  high  m.m.f.  (20  times  as 
great  as  that  of  the  mutual  flux  at  5  per  cent,  exciting  current), 
this  flux  and  the  reactance  voltages  caused  by  it  are  appreci- 


286     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


able,  usually  between  2  per  cent,  and  8  per  cent,  in  modern 
transformers. 

The  distribution  of  the  leakage  flux  between  primary  and 
secondary  winding,  that  is,  between  primary  reactance  x\  and 
secondary  Xz,  is  to  some  extent  arbitrary  (see  discussion  in 
"Theory  and  Calculation  of  Electric  Circuits''))  and  the  methods 
of  test  give  only  the  sum  of  the  primary  and  the  secondary  re- 
actance, the  latter  reduced  to  the  primary  by  the  ratio  of  trans- 
formation :  Xi  +  a2x2. 

116.  The  total  reactance  of  primary  and  secondary,  and  also 


TRANSFORMER 
I  mpedance  and  Short  Circuit  Losses 


7 


.1       .2       .3       .1       .5       .6       .7        .8       .9        1.0      1.1      1.2     1.3      l.i      1.5 

FIG.  156. — Impedance  and  short  circuit  losses  of  transformer. 

the  total  (effective)  resistance  of  primary  and  secondary  winding 
are  measured  by  impressing  voltage  on  the  primary  coil,  with  the 
secondary  winding  short-circuited,  and  measuring  volts,  amperes 
and  watts. 

In  this  test  the  voltage  usually  is  impressed  upon  the  high 
voltage  winding,  as  the  impedance  voltage  is  only  a  small  part 
of  the  operating  voltage  of  the  transformer. 

Such  "impedance  curves"  and  "short-circuit  loss  curves" 
for  the  transformers  in  Figs.  154  and  155  are  shown  in  Fig.  156. 
If  the  short-circuit  loss  is  greater  than  the  sum  of  primary  and 


ALTERNA  TING-CURRENT  TRANSFORMER       287 

secondary  izr  losses,  the  difference  represents  load  losses  caused 
by  eddy  currents  in  the  conductors,  etc.  * 

The  reactance  of  the  transformer  is  often  given  as  percentage. 
Six  per  cent,  reactance  thus  means  that  the  primary  ix,  as  per 
cent,  of  the  primary  impressed  voltage,  plus  the  secondary  ix 
as  per  cent,  of  the  secondary  voltage,  is  6  per  cent.  Or: 

*'*'  +  **'*'=  0.06. 

e\ 

Especially  since  x\  and  xz  cannot  be  separated  experimentally, 
but  the  impedance  test  gives  the  sum  of  primary  reactance  x\, 
and  secondary  reactance  x2  reduced  to  the  primary  by  the  ratio 
of  transformation  a,  that  is 


this  is  permissible. 

The  foremost  effects  of  the  leakage  reactance  of  the  trans- 
former are,  to  affect  the  voltage  regulation,  and  to  determine  the 
short-circuit  current  and  the  mechanical  forces  resulting  from  it. 

117.  The  exciting  current,  being  a  small  and  practically  con- 
stant component  of  the  primary  current,  does  not  affect  the  regu- 
lation of  the  transformer  appreciably,  and  thus  can  be  neglected 
in  the  calculation  of  the  regulation  curve.  If  this  is  done,  the 
secondary  quantities  can  be  reduced  to  the  primary  by  the  ratio 
of  transformation  (or  inversely),  that  is,  by  multiplying  all 
secondary  voltages  and  dividing  all  secondary  currents  by  a,  and 
multiplying  all  secondary  impedances  by  a2,  or  inversely  when 
reducing  from  primary  to  secondary.1 

Or,  primary  and  secondary  impedances  can  be  given  in  per 
cent.,  that  is,  the  primary  ir  and  ix  in  per  cent,  of  the  primary 
voltage,  the  secondary  ir  and  ix  in  per  cent,  of  the  secondary 
voltage,  and  in  this  case,  primary  and  secondary  quantities  can 
be  directly  added.  This  usually  is  the  most  convenient  way,  at 
least  for  approximate  calculation. 

Thus  in  the  transformer  shown  in  Fig.  154,  let 

£  =  0.02  be  the  total  reactance  (2  per  cent.),  at  full  non- 
inductive  load. 

1  As  the  transformation  ratio  of  the  voltage  is  a,  that  of  the  current  is  -i 

the  transformation  ratio  of  the  impedances  (resistance  and  reactance),  is 

volts 

a2,  as  impedance  =  —         — 
amperes 


288     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


p  =  0.02  is  the  total  resistance,  primary  and  secondary  com- 
bined. 

At  the  percentage  p  of  the  non-inductive  load,  the  voltage 
consumed  by  reactance  is  p%  =  0.02  p  and  in  quadrature  with 
the  current  and  thus  with  the  voltage  at  non-inductive  load, 
hence  subtracts  by  ^/difference  of  squares: 


while  the  voltage  consumed  by  the  resistance  is  pp  —  0.02  p 
and  in  phase  with  the  voltage,  hence  directly  subtracts,  leaving: 


-  P2?  ~  PP  =  V  1  -  0.0004  p2-  0.02  p 

as  the  voltage  at  percentage  p  of  load,  given  as  per  cent,  of 
the  open-circuit  or  no-load  voltage.     The  voltage  drop  at-frac- 


REGULATION  of  TRANSFORMER 
Non-inductive  Load; 

I  Z  =  .02  +  .02j 

II  .01  +  .04  j 

III  .01  +  .08  j 


3.5 


.1        .2       .3        .4       .5       .6       .7       .8       .9       1.0      1.1      1.2     1.3      1.4      1.5 

FIG.  157. — Regulation  curve  of  transformer:  non-inductive  load. 

tional  load  p,  as  fraction  of  full-load  voltage,  that  is,  the  regula- 
tion of  the  transformer  at  non-inductiVe  load,  then  is 


R  =  1  -  VI  -  p2^  +  PP  =  1  -  V  1  -  0.0004 p2  +  0.02  p 
or,  resolved  by  the  binomial,  and  dropping  the  higher  terms: 

R  =  PP  +  \  P2?  =  0.02  p  +  0.0002  p2 
=  P  (P  +  \  P?}  =  0.02  p  (1  +  0.01  p) 

As  curves  I,  II,  III  in  Fig.  157  are  shown  the  regulation  curves 
of  three  transformers: 


ALTERNATING-CURRENT  TRANSFORMER       289 


I:  2  per  cent,  resistance  and  2  per  cent,  reactance. 

II :  1  per  cent,  resistance  and  4  per  cent,  reactance. 

Ill :  1  per  cent,  resistance  and  8  per  cent,  reactance. 


FIG.   158. — Vector  diagram  of  transformer  regulation. 


% 

6.5, 

REGULATION  of  TRANSFORMER 
Inductive  Load:     20  Lag 
I      Z  =  .02  +  .02  3 
II             .01  +  .04  3 
III             .01  +  .08  3 

/ 

<<* 

/ 

/ 

5^ 

/ 

/ 

5.0 

/ 

/ 

4^ 

/ 

S 

4.^ 

17 

^^f 

^ 

/ 

x^ 

s^ 

3.0 

/ 

/ 

Ix 

^ 

s^ 

2.5 

/ 

/ 

<> 

^1 

2.0 

/ 

^ 

t> 

^ 

1^ 

S 

/ 

/? 

^ 

1.0 

/ 

£ 

^ 

^ 

X 

X 

Lc 

iad  :— 

—  »- 

.1       ^        .3       .4       ^        .6       .7       .8       .9       1.0     LI     LI     U     1.4     U 

FIG.  159. — Regulation  of  transformer,  moderately  inductive  load. 

Calculated  respectively  by  the  equations  given  at  end  of  next 
paragraph. 

118.  At  inductive  load  of  power-factor  cos  o>,  that  is,  the 
lag  of  the  current  behind  the  voltage  by  angle  w,  the  regulation 


290     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


curve  is  derived  from  the  vector  diagram  Fig.  158.  The  ir 
voltage  is  in  phase  with  the  current,  the  ix  voltage  90  deg.  ahead 
of  the  current.  Resolving  both  of  these  voltages  into  components 
in  phase  and  in  quadrature  with  the  terminal  voltage,  gives 
(Fig.  158): 

16.0 


/ 

% 

/ 

10.0 

REGULATION  of  TRANSFORMER 
Inductive  Load:       60    Lag 
I       Z  =  .02  -f  .02  j 
II              .01+.04J 
III              .01+  .08  j 

/ 

/ 

9.5 

/ 

9.0 

/ 

8.5 

/ 

/ 

8.0 

/ 

7.5 

/ 

7.0 

/ 

/ 

6.5 

III 

/ 

^ 

/ 

/ 

5.5 

/ 

/ 

/ 

/ 

5.0 

/ 

/ 

/ 

4.5 

/ 

^ 

_x 

^^O 

/ 

/ 

/ 

X 

x^ 

3.5 

/ 

/ 

/ 

I 

X 

x 

3.0 

/ 

/ 

/ 

«x 

X 

2.5 

/ 

/ 

/I 

x 

/ 

2.0 

/ 

/ 

.X 

X 

1.5 

/ 

/ 

/ 

^ 

1.0 

/ 

'x 

S 

x^ 

Loa 

d:_ 

—  >- 

.5 

^ 

X 

.1       .2       .3       .4        .6       .6        .7       .8       .9       1.0      1.1      1.2      1.3     1.4      1.5 

FIG.  160. — Regulation  of  transformer,  highly  inductive  load. 

ir  cos  co  and  ix  sin  co  in  phase  with  e, 
ix  cos  co  and  —ir  sin  co  in  quadrature  with  e. 
The  former  thus  directly  subtract,  and  the  latter  subtract  by 
A/difference  of  squares,  thus  giving  as  resultant  voltage : 
—  (ix  cos  co  —  ir  sin  co) 2  —  (ir  cos  co  +  ix  sin  co) 


ALTERNATING-CURRENT  TRANSFORMER       291 


or,  since  ir  at  full  load  as  fraction  of  e  is  p,  and  ix  as  fraction  of 
6  is  £;  at  the  fraction  p  of  the  load:  ir  =  pp,  ix  =  p£,  the  re- 
sultant voltage  is: 


\/l  —  p2  (^  cos  co  —  p  sin  co)2  —  P  (p  cos  co  +  £  sin  co) 

and  the  regulation  of  the  transformer,  at  inductive  load  of 
angle  of  lag  co,  thus  is- 

R  =  1  —  A/1  —  p2  (£  cos  co  —  p  sin  co)2  +   p  (p  cos   co  +  £  sin  co). 

Resolving  again  the  square  root  by  the  binomial,  and  arrang- 
ing, gives,  by  dropping  out  terms  of  higher  order: 

v2 
R  —  p  (p  cos  co  +  £  sin  co)  +  ~  (£  cos  co  —  p  sin  co)2 

In  Figs.  159  and  160  are  shown,  for  the  angles  of  lag  co  =  20° 
(moderately  inductive  load,  94  per  cent,  power-factor),  and 
co  =  60°  (highly  inductive  load,  50  per  cent,  power-factor),  the 
regulation  of  the  same  three  transformers  as  in  Fig.  157,  cal- 
culated respectively  from  the  expression: 

REGULATION  OF  TRANSFORMERS 


Per  cent,  resistance,  p  = 

0.02 

0.01 

0.01 

Per  cent,  reactance,  £  = 

0.02 

0.04 

0.08 

^  w  of  lag: 

Curve  I 

Curve  II 

Curve  III 

Fig.  157,  0°             R  = 

0.02p 

O.Olp 

O.Olp 

(1  +0.01p) 

(1  +0.08p) 

(1  +0.32p) 

Fig.  159,  20°           R  = 

0.0256p 

0.0231p 

0.0368p 

(1  +  0.0027p) 

(1  +  0.025p) 

(1  +0.07?) 

Fig.  160,  60°           R  = 

0.0273p 

0.0396p 

0.0743p 

(!4+0.001p) 

(1  +  0.0016p) 

(1+0.  0066  p) 

Non-inductive 

20°  lag. 

60°  lag. 

P 

I 

II 

III 

I 

II 

III 

I 

II 

III 

0.2 

0.40 

0.20 

0.21 

0.51 

0.46 

0.75 

0.55 

0.79 

1.49 

0.4 

0.80 

0.41 

0.45 

1.02 

0.93 

1.52 

1.09 

1.58 

2.99 

0.6 

1.20 

0.63 

0.71 

1.54 

1.40 

2.30 

1.64 

2.38 

4.48 

0.8 

1.61 

0.85 

1.00 

2.05 

1.88 

3.13 

2.18 

3.17 

5.98 

1.0 

2.02 

1.08 

1.32 

2.56 

2.37 

3.94 

2.73 

3.96 

7.48 

1.2 

2.43 

1.31 

1.66 

3.08 

2.85 

4.79 

3.28 

4.76 

9.00 

1.4 

2.84 

1.56 

2.03 

3.59 

3.34 

5.67 

3.83 

5.56 

10.50 

1.6 

3.25 

1.80 

2.42 

4.10 

3.83 

6.56 

4.38 

6.35 

12.00 

i 

292     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


119.  As  seen,  at  non-inductive  load,  Fig.  157,  the  reactance  of 
the  transformer,  even  if  fairly  high,  has  practically  no  effect,  but 
the  resistance  controls  the  regulation. 

At  moderately  inductive  load  reactance  as  well  as  resistance 
affect  the  regulation;  doubling  the  reactance  while  halving  the 
resistance,  gives  practically  the  same  regulation. 


FIG.  161. — High  reactance  transformer  construction. 

At  highly  inductive  load  the  reactance  of  the  transformer  be- 
gins to  predominate  over  the  resistance  in  affecting  the  regulation. 

Thus,  where  close  regulation  is  required,  as  in  lighting  and 
general  distribution  transformers,  low  reactance  is  of  impor- 
tance. This  is  given  by  reducing  the  section  of  the  leakage  path — 
that  is,  bringing  primary  and  secondary  windings  as  close  together 


FIG.  162. — Low  reactance  transformer  construction. 

as  possible — and  by  reducing  the  m.m.f.  which  produces  the  leak- 
age flux,  by  subdividing  primary  and  secondary  winding  into  a 
number  of  coils  and  intermixing  these  coils,  so  that  the  leakage 
flux  of  each  path  is  due  to  a  small  part  of  the  total  m.m.f.  of 
primary  or  secondary  only,  as  shown  in  Figs.  161  and  162.  In 
Fig.  162  the  m.m.f.  of  each  of  the  four  leakage  paths  is  due  to 


ALTERNATING-CURRENT  TRANSFORMER       293 

one-fourth  of  the  m.m.f.  as  in  Fig.  161,  and  the  leakage  flux 
density  thus  reduced  to  one-fourth  of  what  it  is  in  Fig.  161. 
As  furthermore  the  section  of  each  leakage  flux  in  Fig.  162  is 
materially  less  than  in  Fig.  161,  due  to  the  lesser  thickness  of 
the  coils,  it  follows  that  in  Fig.  162  the  leakage  flux  interlinked 
with  each  turn  of  each  winding,  and  thus  the  reactance  of  the 
transformer,  is  materially  less  than  one-quarter  of  what  it  is  in 
Fig.  161. 

The  regulation  of  the  transformer  at  anti-inductive  load,  that 
is,  for  leading  secondary  current,  obviously  is  given  by  the  same 
equation  as  that  for  lagging  current,  by  merely  substituting 
—  co  for  co. 

V.  Short-circuit  Current 

120.  If  a  short  circuit  occurs  at  the  secondary  terminals  of  a 
transformer,  and  the  power  supply  at  the  primary  is  sufficient  to 
maintain  the  primary  terminal  voltage,  the  primary  and  second- 
ary currents  of  the  transformer  are  limited  by  its  impedance  only. 
Thus,  if 

r  =  P  +  j* 

is  the  impedance  voltage,  as  fraction  of  full-load  voltage,  the  short- 
circuit  current  of  the  transformer  is 

1  1 

of  the  full-load  current,  thus  usually  is  very  large.  In  the  three 
instances  illustrated  in  Figs.  157,  159  and  160,  with 

f  =  0.02  +  0.02  j,  hence        f  =0.028 

0.01  +  0.04  j  0.04 

0.01  +  0.08  j  0.08 

the  short-circuit  current  thus  is  36,  25  and  12.5  times  full-load 
current,  respectively. 

As  seen,  with  the  exception  of  very  low  reactance  transformers, 
it  is  essentially  the  reactance  which  determines  the  total  im- 
pedance and  thus  the  short-circuit  current. 

121.  Primary  current  and  secondary  current  in  the  trans- 
former, being  opposite  in  phase,  repel  each  other.     This  repul- 
sion is  proportional  to  the  product  of  primary  and  secondary 
current,   thus,   since  primary  and  secondary  current  are   (ap- 

19 


294     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

proximately)  proportional  to  each  other,  the  repulsion  is  pro- 
portional to  the  square  of  the  current.  The  repulsion  is  small 
at  full  load,  but  in  low-reactance  transformers,  with  'short-circuit 
currents  from  forty  to  fifty  times  full-load  current,  the  mechanical 
forces  have  increased  1600  to  2500  fold,  and  then,  with  large 
power  transformers,  reach  formidable  values,  amounting  to  many 
hundred  tons,  and  then  it  is  economically  difficult  to  build  trans- 
formers with  the  coils  supported  so  rigidly  as  to  stand  such  forces. 
Thus  far  very  few  generating  systems  exist  of  such  large  size 
as  to  be  capable  of  maintaining  full  voltage  at  the  primary  ter- 
minals of  a  large  transformer  at  secondary  short  circuit,  but  their 
number  is  increasing,  and  thus  the  necessity  of  limiting  the  short- 
circuit  current  of  large  power  transformers  to  a  mechanically 
safe  value  is  becoming  increasingly  important.  This  means  a 
construction  providing  for  considerable  internal  reactance.  As 
the  regulation  of  large  power  transformers  is  of  no  serious  impor- 
tance, the  desirability  of  low  reactance,  which  exists  in  the  small 
lighting  and  general  distribution  transformers,  does  not  exist  in 
large  power  transformers,  and  modern  practice  tends  toward  the 
use  of  internal  reactance  of  4  to  8  per  cent.,  to  secure  reasonable 
mechanical  safety. 

VI.  Heating  and  Ventilation 

122.  As  the  transformer  is  a  stationary  apparatus,  it  does  not 
have  the  advantage  of  dissipating  the  heat  produced  by  the 
internal  losses,  by  the  natural  ventilation  of  the  air  currents  pro- 
duced by  the  centrifugal  forces  in  rotating  apparatus,  and  it  is 
therefore  fortunate  that  the  transformer  is  the  most  efficient 
apparatus  (except  perhaps  the  electrostatic  condenser)  and  thus 
has  to  dissipate  less  heat  than  any  other  apparatus  of  the  same 
output.  Thus  in  smaller  transformers  radiation  and  the  natural 
convection  from  the  surface  are  often  sufficient  to  keep  the  tem- 
perature within  safe  limits. 

Smaller  distribution  transformers  usually  are  installed  out- 
doors, on  poles,  and  then  require  protection  by  enclosure  in  an 
iron  case  or  tank.  This  still  further  reduces  the  heat  radiation, 
and  therefore  such  transformer  cases  are  now  almost  always  filled 
with  oil,  the  oil  serving  to  carry  the  heat  from  the  transformer 
iron  and  windings  to  the  case.  Incidentally,  the  oil  filling  also 
protects  the  transformer  from  the  failure  of  insulation  by  con- 


ALTERNATING-CURRENT  TRANSFORMER       295 

densation  of  moisture  during  the  variation  of  atmospheric  tem- 
perature and  humidity. 

In  larger  oil-cooled  transformers,  the  tank  is  made  corrugated, 
even  with  large  double  corrugations,  to  give  a  very  large  external 
surface  to  dissipate  the  heat. 

Much  more  effectively,  however,  the  heat  can  be  carried 
away  by  mechanical  ventilation,  and  size  and  cost  of  the  trans- 
former thereby  materially  reduced.  Therefore  practically  all 
larger  transformers  have  forced  ventilation.  Various  methods 
of  forced  ventilation  are: 

(a)  Oil  circulation.  The  warm  oil  is  pumped  from  the  top 
of  the  transformer  tank,  through  some  cooling  device.  Often 
also  a  drying  device  to  take  out  any  trace  of  moisture — and  then 
fed  back  into  the  bottom  of  the  tank. 

(6)  Water  circulation.  Cooling  water  is  pumped  through 
a  system  of  pipes  located  under  the  oil  at  the  top  of  the  trans- 
former tank.  This  is  the  most  common  design  of  large  trans- 
formers. 

(c)  Air  blast.  Coils  and  iron  are  subdivided  by  ventilating 
ducts,  and  a  low-pressure  air  blast  forced  through  the  ventilating 
ducts.  This  is  the  cleanest  method,  as  no  oil  is  used.  However, 
it  is  limited  to  low  and  moderate  voltages — up  to  about  33,000; 
at  higher  voltage,  the  mechanical  and  chemical  action  of  corona 
appearing  at  the  coils  reduces  their  life,  and  the  oil  becomes 
necessary  for  insulation. 

Numerous  modifications  of  the  various  types  have  been 
built  and  are  in  use,  as  water-cooled  oil  transformers  with 
natural  circulation  of  the  water  through  outside  radiating 
pipes,  etc. 

VH.  Types  of  Transformers 

123.  As  the  transformer  consists  of  a  magnetic  circuit  inter- 
linked with  two  electric  circuits,  two  constructive  arrangements 
are  possible :  The  electric  circuits  may  be  inside,  and  surrounded 
by  the  magnetic  circuit  as  shell,  shell-type  transformer;  or  the 
magnetic  circuit  may  be  arranged  inside,  as  core,  and  sur- 
rounded by  the  electric  circuits,  core-type  transformer. 

In  their  simplest  form,  Fig.  163  shows  diagrammatically 
the  core-type  transformer,  with  the  iron  Fe  as  inside  circular 
core,  built  up  of  laminations  or  of  iron  wire,  and  the  windings 
Cu  outside;  Fig.  164  shows  diagrammatically  the  shell-type 


296     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

transformer,  with  the  copper  windings  inside,  as  Cu,  and  the 
iron  shell  Fe  wound  around  it,  of  iron  wire,  etc.  However,  the 
circular  form  163  is  used  to  a  limited  extent  only,  in  small  trans- 
formers, autotransformers  and  reactances,  and  the  form  164 
practically  never  used,  and  in  the  constructive  modification  from 
these  diagrammatic  types,  it  is  often  difficult  to  decide  to  which 
type  to  assign  the  transformer. 


FIG.  163. — Diagram  of  core  type  transformer. 


FIG.   166. — Diagram  of  shell  type  transformer. 

The  typical  shell-type  transformer  of  today  is  shown  in 
section  in  Fig.  165,  with  the  magnetic  circuit  Fe,  and  the  high 
voltage  windings  P  and  low- voltage  windings  S  intermixed  with 
each  other. 

Core-type  transformers  are  shown  in  section  in  Figs.  166 
and  167,  the  former  with  one,  the  latter  with  two  cores,  and 
with  two  different  coil  arrangements,  the  intermixed  and  the 
concentric. 


ALTERNATING-CURRENT  TRANSFORMER       297 


For  the  transformation  of  three-phase  circuits,  three  separate 
single-phase  transformers  may  be  used,  and  their  primaries  and 


FIG.  165. — Shell  type  transformer. 


FIG.    166. — Single-coil  core  type 
transformer. 


FIG.     167. — Two  coil  core  type 
transformer. 


FIG.  168. — Shell  type  three-phase      FIG.  169. — Core  type  three-phase 
transformer  diagram.  transformer  diagram. 

secondaries  then  connected  in  ring  or  delta  connection  or  in  star 
or  Y  connection,  giving  the  four  arrangements: 

AA,     AF,     FA,     YY. 

Or  two  transformers  may  be  used,  arranged  in  T  connection 
or  in  open  A  connection,  as  further  discussed  under  three-phase 
systems.  Or  a  three-phase  transformer  may  be  used.  Diagram- 


298     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

matically,  the  three-phase  transformer  can  be  represented  by 
Fig.  168,  shell  type,  and  Fig.  169,  core  type. 

124.  While  in  its  magnetic  and  electrical  characteristics 
there  is  no  essential  difference  between  the  single-phase  shell- 
type  and  the  single-phase  core-type  transformer,  there  is  a 
material  difference  in  the  three-phase  transformer.  In  the  shell 
type,  Fig.  168,  a  short  circuit  of  one  of  the  three  phases  does  not 
affect  the  magnetic  and  thus  the  electric  circuit  of  the  other 
two  phases,  in  the  core  type  Fig.  169,  however,  a  short  circuit  of 
one  of  the  three  phases  short  circuits  the  magnetic  return  of 
the  other  two  phases,  and  so  acts  as  a  partial  electrical  short 
circuit  of  these  two  other  phases. 

In  shell-type  transformers,  Fig.  168,  a  triple  harmonic  of 
flux  can  exist,  but  not  in  the  core  type,  Fig.  169.  In  the  three- 


FIG.  170. — Shell  type  three-phase  transformer. 

phase  system,  the  three  voltages,  currents,  etc.,  are  displaced 
in  phase  from  each  other  by  120°.  Their  third  harmonics 
therefore  are  displaced  in  phase  from  each  other  by  3  X  120°, 
that  is,  by  360°,  or  in  other  words,  are  in  phase  with  each  other. 
In  Fig.  169,  such  triple  frequency  fluxes  in  the  three  cores  would 
have  no  magnetic  return,  except  by  leakage  through  the  air, 
that  is,  cannot  exist,  except  in  negligible  intensity,  and  there- 
fore the  core  type  of  three-phase  transformer  cannot  give  any 
serious  triple  frequency  voltage.  In  the  shell  type  Fig.  168, 
however,  the  three  triple  frequency  fluxes,  being  in  phase  with 
each  other,  produce  a  triple  frequency  single-phase  flux  through  a 
closed  magnetic  circuit.  Where  the  circuit  conditions  and 
connections  are  such  as  to  give  a  triple  harmonic — as  with  YY 
connection — the  shell-type  three-phase  transformer  may  produce 
triple  frequency  voltages,  resulting  from  the  triple  frequency 


ALTERNATING-CURRENT  TRANSFORMER       299 

flux,  and  under  unfavorable  conditions,  as  when  connecting  to  a 
system  of  high  capacity — which  intensifies  these  voltages — 
this  may  lead  to  destructive  voltages,  and  YY  connections 
with  shell-type  three-phase  transformers  thus  lead  to  serious  high 
voltage  dangers. 

125.  The  usual  shell-type  construction  of  three-phase  trans- 
formers is  shown  in  section  in  Fig.  170,  the  core  type  in  Fig. 
171. 

In  Fig.  170  economy  requires  that  the  middle  phase  is  con- 
nected in  opposite  direction  to  the  outside  phases,  so  that  the 
iron  between  the  successive  phases,  at  1,  2  and  2,  3,  carries  the 
sum  of  two  of  the  three-phase  fluxes,  which,  as  the  fluxes  are 
120  deg.  apart,  equals  one  of  the  fluxes.  If  the  middle  phase 
were  not  reversed,  1,  2  and  2,  3  would  carry  the  difference  of 


II 

II 

FIG.  171. — Core  type  three-phase  transformer. 

two  fluxes  120  deg.  apart,  and  this  difference  is  V3  times  each 
flux,  thus  would  give  a  much  higher  loss. 

In  Fig.  171  usually  the  exciting  current  of  the  middle  phase 
is  somewhat  less  than  that  of  the  outside  phase,  since  the  magnetic 
reluctance  of  the  middle  phase  is  slightly  lower. 

VIII.  Autotransformer 

126.  If  in  a  transformer  a  part  of  the  secondary  winding  is 
used  as  primary,  or  inversely,  the  transformer  is  an  autotrans- 
former,  sometimes  also  called  compensator. 

Thus  let  in  a  transformer  Fig.  172  primary  current,  voltage  and 
turns  be  respectively  ii,  e^  ni,  and  secondary  current,  voltage  and 

turns   be   t'2,    e2,  n2,  thus  the  ratio  of  transformation  a  =  —  • 

n2 

Assuming  HI  >  nz,  then  in  any  n2  of  the  HI  primary  turns,  the  same 
voltage  is  induced  as  in  the  n2  secondary  turns,  and  we  could  thus 


300     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


use  any  n2  primary  turns  as  secondary  turns,  provided  we  make 
them  of  sufficient  copper  section  to  carry  the  secondary  current. 

The  n2  turns  in  Fig.  173  thus  are  in  common  to  primary  and 
secondary  circuit.  As  primary  and  secondary  current  are  (ap- 
proximately) opposite  in  phase,  the  current  in  the  common  turns 
of  Fig.  173  is  (approximately,  that  is,  neglecting  exciting  current) 
the  difference  between  secondary  and  primary  current,  i2  —ii, 
thus  less  than  the  secondary  current  i'2,  and  as  the  result,  the  com- 
mon turns  in  Fig.  173  may  be  made  of  less  copper  section  than  the 
secondary  turns  in  Fig.  172,  while  the  number  of  primary  turns 
is  reduced  by  n2.  Thus  an  autotransformer  requires  less  copper, 
that  is,  is  smaller  and  cheaper  than  a  transformer  of  the  same 
output. 

127.  In  the  transformer  Fig.  172,  the  size  is  determined  by  the 
number  of  turns  and  turn  sections,  that  is,  by  e\  X  ii  +  £2  X  iz 


FIG.    172. — Diagram   of  trans- 
former. 


FIG.    173. — Diagram   of   auto- 
transformer. 


(the  turns  being  proportional  to  the  voltage,  the  turn  section  to 
the    current,    the    same    magnetic    flux    assumed).     But    since 

61  =  aez  and  i\  =  — ,  e\i\  =  e2i2,  and  the  size  of  the  transformer 

Fig.  172  thus  is  proportional  to  2  e-#2,  that  is,  to  2  P,  or  twice  the 
output. 

In  the  autotransformer  Fig.  173,  the  nz  common  turns  are  tra- 
versed by  the  difference  of  secondary  and  primary  current,  at 
secondary  voltage,  and  the  size  of  this  common  part  of  the  wind- 
ing thus  is:  62  (iz  —  ii).  The  remaining  part  of  the  winding, 
of  n\  —  n2  turns,  that  is,  of  voltage  e\  —  e2)  is  traversed  by  the 
primary  current  ii,  hence  of  size  i\  (e\  —  e2),  and  the  total  size 
of  the  autotransformer  thus  is : 

62  (*2  —  ii)  +  i\  (e\  —  e2) 


ALTERNATING-CURRENT  TRANSFORMER       301 

but,  substituting  again  for  ii  and  ei,  gives  as  the  size  of  the  auto- 
transformer: 


(ae2  -  es)  =  2 

-"•('-3 

hence,  the  ratio  of  size  of  autotransformer  and  of  transformer 
of  the  same  output,  is: 

_  autotransformer  _  1. 

transformer  a 

If  the  ratio  a  =  2,  as  transforming  between  115  and  230  volts, 
7  =  J£,  that  is,  the  autotransformer  has  half  the  size  of  the  trans- 
former, or,  more  correctly  stated,  the  autotransformer  is  as  large 
as  a  transformer  of  half  the  output. 

If  the  ratio  a  =  1.1,  as  raising  (or  lowering)  the  voltage  10 
per  cent,  by  autotransformer,  this  autotransformer  has  the  size 
7  =  0.1  that  is,  is  as  small  as  a  transformer  of  one-tenth  the  out- 
put. 

If  the  ratio  a  =  10,  as  transforming  between  2300  and  230, 
7  =  0.9,  that  is,  the  autotransformer  is  only  10  per  cent,  smaller 
than  the  transformer. 

The  saving  in  size  —  and  therewith  in  efficiency  and  cost  —  by 
the  use  of  the  autotransformer  thus  is  the  greater,  the  lower  the 
transformation  ratio  a,  but  becomes  negligible  at  high  trans- 
formation ratios.  Thus  autotransf  ormers  are  very  economical  for 
use  in  moderate  voltage  transformation,  as  a  voltage  change  by 
10  or  20  per  cent.,  or  even  for  doubling  the  voltage,  or  dividing 
it  in  two,  but  not  for  high  voltage  ratios. 

128.  The  most  serious  disadvantage  of  the  autotransformer 
obviously  is  that  it  electrically  interconnects  primary  and  sec- 
ondary circuit  and  thereby  puts  the  voltage  of  the  higher  voltage 
circuit  onto  the  lower  voltage  circuit.  Thus,  when  using  auto- 
transformers,  the  insulation  of  the  low  voltage  circuit  and  the 
high  potential  tests  of  all  the  apparatus  used  in  the  low  voltage 
circuit  must  be  those  of  the  high  voltage  circuit.  Furthermore, 
a  ground  in  one  of  the  two  circuits  of  an  autotransformer  also 
is  a  ground  on  the  other  circuit,  while  with  a  transformer,  a 
ground  on  the  secondary  does  not  ground  the  primary,  and  in- 
versely. With  low  voltages,  as  115  -5-  230  volt  transformation, 
this  is  usually  of  no  importance.  It  would  be  a  serious  objection 


302     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

when  attempting  the  use  of  autotransformers  between  2300 
and  230  volts.  For  instance,  a  ground  at  the  off  side  of  the  high- 
voltage  winding,  at  A  in  Fig.  173,  would  put  the  entire  secondary 
winding  2300  to  2070  volts  above  ground,  and  thus  the  secondary 
circuit  would  kill  anybody  who  touches  it  while  standing  on  the 
ground. 

Any  transformer  of  voltage  e\  and  62  and  currents  i\,i%  can  be 
converted  into  an  autotransformer,  by  connecting  primary  and 
secondary  in  series,  of  voltages  e\  +  ez  and  ez  and  currents  i\ 
and  iz  +  i\.  And  inversely,  any  autotransformer,  by  disconnect- 
ing the  two  sections  of  the  coil,  would  give  (provided  that  the 
insulation  is  sufficient)  a  transformer  of  (ei  —  e^)  X  i\  primary, 
and  ez  X  (i*  —  ii)  secondary  circuit. 

The  regulation  of  an  autotransformer  is  better,  and  the  effi- 
ciency higher,  than  that  of  the  same  structure  as  transformer, 
and  the  per  cent,  reactance  lower,  that  is  the  short-circuit  current 
higher  in  the  autotransformer  than  in  the  same  structure  as 
transformer.  Very  often  it  is  difficult  to  build  autotransformers 
with  sufficiently  high  internal  reactance,  to  make  them  safe 
under  momentary  short  circuit  as  autotransformers,  while  they 
may  be .  perfectly  safe  as  transformers,  where  the  reactance  is 
higher.  This  is  a  serious  objection  to  the  use  of  autotransformers 
in  high-power  systems. 

IX.  Reactors 

(Reactive  Coils,  Reactances) 

129.  The  reactor  consists  of  one  electric  circuit  interlinked 
with  a  magnetic  circuit,  and  its  purpose  is,  not  to  transform 
power,  but  to  produce  wattless  or  reactive  power,  that  is,  lagging 
current,  or  what  amounts  to  the  same,  leading  voltage.  While 
therefore  theoretically  we  cannot  speak  of  an  ''efficiency"  of  a 
reactor,  since  there  is  no  power  output,  nevertheless  in  the  in- 
dustry the  expression  " efficiency  of  a  reactive  coil"  is  gener- 
ally used,  and  generally  understood,  in  the  conventional  definition : 

T^C     •  1°SS 

Efficiency  =  1  —  -. — 

input 

and  the  input  is  given  in  total  volt-amperes,  the  loss  in  energy 
volt-amperes,  that  is,  watts.  The  efficiency  then  is  1  —  power- 
factor. 


ALTERNATING-CURRENT  TRANSFORMER       303 

The  transformer  at  open  circuit  is  a  reactor,  but  a  very  poor 
one,  as  its  power-factor  is  high,  that  is,  the  efficiency  low. 

In  the  transformer,  the  exciting  ampere-turns  are  the  (vector) 
difference  between  primary  and  secondary  ampere-turns,  are 
wasted,  and  therefore  made  as  low  as  possible,  by  using  a  closed 
magnetic  circuit.  In  the  reactor,  no  secondary  circuit  exists, 
but  the  exciting  ampere-turns  are  the  purpose  of  the  device, 
thus  should  be  as  large  as  possible.  That  is,  to  convert  a  trans- 
former into  a  reactor,  the  reluctance  of  the  magnetic  circuit 
must  be  increased  so  as  to  make  the  exciting  ampere-turns 
equal  to  the  total  full-load  ampere-turns  of  the  structure  as 
transformer.  This  is  done  by  inserting  an  air  gap  into  the 
magnetic  circuit.  Such  a  gap  may  be  either  a  single  gap,  or  a 
number  of  smaller  air  gaps,  or  one  or  a  number  of  slots  cutting 
almost  through  the  magnetic  circuit,  but  leaving  narrow  bridges, 


FIG.  174. — Bridged  air-gap  reactor. 

as  shown  in  Fig.  174.  This  latter  offers  the  advantage  of  a 
better  mechanical  structure,  less  liability  to  noise  and  to  magnetic 
leakage,  but  when  used  in  series  in  high  voltage  circuits,  may 
lead  to  voltage  peaks  at  the  moment  of  current  reversal,  which 
may  endanger  the  insulation.  The  use  of  a  number  of  small 
air  gaps  instead  of  one  large  one  distributes  the  magnetic  leakage 
and  thus  gives  less  liability  to  eddy  currents  in  the  conductors. 

130.  A  transformer  of  output  P  =  e2iz  has  a  size  of  winding 
space  of  ezi2  +  #iii  =  2  e2z*2,  that  is  (with  the  air  gap  inserted 
into  the  magnetic  circuit),  gives  a  reactor  of  the  capacity  ei  =  2  P. 
That  is,  a  reactor  has  the  size  of  a  transformer  of  half  its  output. 

Reactors  are  frequently  used  in  series  to  apparatus,  and  the  vol- 
tage consumed  by  the  reactance  then  varies  with  the  current,  and 
is,  due  to  the  air  gap,  proportional  to  the  current  up  to  the  value 
where  the  iron  part  of  the  reactance  begins  to  saturate,  as  shown 
by  the  characteristic  curve  of  a  reactance,  Fig.  175,  the  "volt- 


304     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


ampere  characteristic."  Then  the  voltage  increases  less  than 
proportional  to  the  current,  or  inversely,  the  current  increases 
out  of  proportion  to  the  voltage,  that  is,  the  reactance  decreases 
and  wave-shape  distortion  occurs.  Reactances  thus  must  be 
designed  so  that  at  the  highest  currents  (or  voltages),  at  which 
they  may  be  called  upon  to  develop  their  reactance,  their  magnetic 
circuit  is  still  below  saturation. 

Industrially,  reactors  are  often  denoted  in  per  cent.     Thus  for 


Volt- 


REACTOR 
Ampere  Characteristic 


Rec 


Vo  ts  e 


FIG.  175. — Volt-ampere  characteristic  of  reactor. 

phase  control  in  synchronous  converter  circuits,  15  per  cent,  re- 
actances are  used.  This  means,  at  full-load  current,  the  voltage 
consumed  by  these  reactances  is  15  per  cent,  of  the  circuit  voltage. 
131.  With  the  increasing  size  and  increasing  voltage  of  modern 
central  stations  and  the  use  of  high-speed  turbo-alternators  ca- 
pable of  momentarily  giving  very  high  short-circuit  currents,  the 
amount  of  power,  which  can  be  developed  momentarily  by  a  short 
circuit  in  the  system  near  the  generating  station,  has  reached  such 


ALTERNATING-CURRENT  TRANSFORMER       305 

destructive  values,  that  a  limitation  of  this  power  has  become 
necessary,  and  as  economy  of  operation  forbids  sectionalizing  the 
system  into  a  number  of  smaller  units,  this  has  led  to  the  exten- 
sive use  of  power-limiting  reactances,  in  the  generator  leads,  in 
the  bus  bars,  tie  feeders  and  even  the  power  feeders.  Such  re- 
actances are  used  of  2'  to  8  per  cent.,  and  in  bus  bars  even  up  to 
25  per  cent.,  and  in  case  of  a  local  short  circuit,  limit  the  current 
which  can  flow.  Thus  a  4  per  cent,  reactance  would  at  a  short 

circuit  just  beyond  the  reactance  limit  the  current  to  -r  -  = 

4  per  cent. 

25  times  the  normal,  etc.  But  to  do  so,  the  reactance  must  still 
be  there  at  twenty-five  times  its  rated  current,  that  is,  when  ab- 
sorbing full  circuit  voltage  instead  of  its  normal  4  per  cent, 
thereof.  If  then  iron  is  used  in  the  magnetic  circuit  of  such  a 
reactance,  the  density  must  be  so  low,  that  at  twenty-five  times 
this  density  (or  at  12.5  times  with  an  8  per  cent,  reactance,  etc.), 
it  does  not  yet  saturate.  When  limited  to  such  very  low  mag- 
netic densities  in  the  iron,  the  mass  of  iron  becomes  so  enormous, 
that  it  becomes  more  economical  to  use  an  air  circuit  throughout. 
Reactances,  which  must  retain  their  reactance,  that  is,  must 
not  saturate  at  many  times  their  normal  current,  such  as  power 
limiting  reactances,  thus  are  built  without  iron  in  the  magnetic 
circuit. 


E.  INDUCTION  MACHINES 
I.  General 

132.  The  direction  of  rotation  of  a  direct-current  motor, 
whether  shunt-  or  series-wound,  is  independent  of  the  direction  of 
the  current  supplied  thereto;  that  is,  when  reversing  the  current 
in  a  direct-current  motor  the  direction  of  rotation  remains  the 
same.  Thus  theoretically  any  continuous-current  motor  should 
operate  also  with  alternating  currents.  Obviously  in  this  case 
not  only  the  armature  but  also  the  magnetic  field  of  the  motor 
must  be  thoroughly  laminated  to  exclude  eddy  currents,  and  care 
taken  that  the  currents  in  the  field  and  armature  circuits  reverse 
simultaneously.  Obviously  the  simplest  way  of  fulfilling  the 
latter  condition  is  to  connect  the  field  and  armature  circuits  in 
series  as  alternating-current  series  motor.  Such  motors  are  used 
to  a  considerable  extent,  but,  like  the  shunt  motor,  have  the  dis- 
advantage of  a  commutator  carrying  alternating  currents.  • 

The  shunt  motor  on  an  alternating-current  circuit  has  the 
objection  that  in  the  armature  winding  the  current  should  be 
power  current,  thus  in  phas£  with  the  e.m.f.,  while  in  the  field 
winding  the  current  is  lagging  nearly  90  deg.,  as  magnetizing 
current.  Thus  field  and  armature  would  be  out  of  phase  with 
each  other.  To  overcome  this  objection  either  there  is  inserted 
in  series  with  the  field  circuit  a  condenser  of  such  capacity  as  to 
bring  the  current  back  into  p>hase  with  the  voltage,  or  the  field 
may  be  excited  from  a  separate  e.m.f.  differing  90  deg.  in  phase 
from  that  supplied  to  the  armature.  The  former  arrange- 
ment has  the  disadvantage  of  requiring  almost  perfect  con- 
stancy of  frequency,  and  therefore  is  not  practicable.  In  the 
latter  arrangement  the  armature  winding  of  the  motor  is  fed  by 
one,  the  field  winding  by  the  other  phase  of  a  quarter-phase  sys- 
tem, and  thus  the  current  in  the  armature  brought  approximately 
into  phase  with  the  magnetic  flux  of  the  field. 

Such  an  arrangement  obviously  loads  the  two  phases  of  the 
system  unsymmetrically,  the  one  with  the  armature  power 
current,  the  other  with  the  lagging  field  current.  To  balance 
the  system  two  such  motors  may  be  used  simultaneously  and 

306 


INDUCTION  MACHINES  307 

combined  in  one  structure,  the  one  receiving  power  current  from 
the  first,  magnetizing  current  from  the  second  phase,  the  second 
motor  receiving  magnetizing  current  from  the  first  and  power 
current  from  the  second  phase. 

The  objection  that  the  use  of  the  commutator  is  complicated 
and  greatly  limits  the  design  to  avoid  serious  sparking  can  be  en- 
tirely overcome  by  utilizing  the  alternating  feature  of  the  current ; 
that  is,  instead  of  leading  the  current  into  the  armature  by  com- 
mutator and  brushes,  producing  it  therein  by  electromagnetic 
induction,  by  closing  the  armature  conductors  upon  themselves 
and  surrounding  the  armature  by  a  primary  coil  at  right  angles  to 
the  field  exciting  coil. 

Such  motors  have  been  built,  consisting  of  two  structures  each 
containing  a  magnetizing  circuit  acted  upon  by  one  phase  and  a 
primary  power  circuit  acting  upon  a  closed-circuit  armature  as 
secondary  and  excited  by  the  other,  phase  of  a  quarter-phase 
system  (Stanley  motor) . 

Going  still  a  step  further,  the  two  structures  can  be  com- 
bined into  one  by  having  each  of  the  two  coils  fulfill  the  double 
function  of  magnetizing  the  field  and  producing  currents  in  the 
secondary  which  are  acted  upon  by  the  magnetization  produced 
by  the  other  phase. 

Obviously,  instead  of  two  phases  in  quadrature  any  number  of 
phases  can  be  used. 

This  leads  us  by  gradual  steps  of  development  from  the  con- 
tinuous-current shunt  motor  to  the  alternating-current  polyphase 
induction  motor. 

In  its  general  behavior  the  alternating-current  induction  motor 
is  therefore  analogous  to  the  continuous-current  shunt  motor. 
Like  the  shunt  motor,  it  operates  at  approximately  constant  mag- 
netic density.  It  runs  at  fairly  constant  speed,  slowing  down 
gradually  with  increasing  load.  The  main  difference  is  that  in 
the  induction  motor  the  current  in  the  armature  does  not  pass 
through  a  system  of  brushes,  as  in  the  continuous-current  shunt 
motor,  but  is  produced  in  the  armature  as  the  short-circuited 
secondary  of  a  transformer;  and  in  consequence  thereof  the 
primary  circuit  of  the  induction  motor  fulfills  the  double  func- 
tion of  an  exciting  circuit  corresponding  to  the  field  circuit  of 
the  continuous-current  machine  and  a  primary  circuit  produc- 
ing a  secondary  current  in  the  secondary  by  electromagnetic 
induction. 


308     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

133.  Since  in  the  secondary  of  the  induction  motor  the  cur- 
rents are  producjed  by  induction  from  the  primary  impressed 
currents,  the  induction  motor  in  its  electromagnetic  features  is 
essentially  a  transformer;  that  is,  it  consists  of  a  magnetic  cir- 
cuit or  magnetic  circuits  interlinked  with  two  electric  circuits 
or  sets  of  circuits,  the  primary  and  the  secondary  circuits.  The 
difference  between  transformer  and  induction  motor  is  that  in 
the  former  the  secondary  is  fixed  regarding  the  primary,  and  the 
electric  energy  in  the  secondary  is  made  use  of,  while  in  the  latter 
the  secondary  is  movable  regarding  the  primary,  and  the  me- 
chanical force  acting  between  primary  and  secondary  is  used. 
In  consequence  thereof  the  frequency  of  the  currents  in  the  sec- 
ondary of  the  induction  motor  differs  from,  and  as  a  rule  is  very 
much  lower  than,  that  of  the  currents  impressed  upon  the  pri- 
mary, and  thus  the  ratio  of  e.m.fs.  generated  in  primary  and  in 
secondary  is  not  the  ratio  of  their  respective  turns,  but  is  the  ratio 
of  the  product  of  turns  and  frequency. 

Taking  due  consideration  of  this  difference  of  frequency  be- 
tween primary  and  secondary,  the  theoretical  investigation  of  the 
induction  motor  corresponds  to  that  of  the  stationary  trans- 
former. The  transformer  feature  of  the  induction  motor  pre- 
dominates to  such  an  extent  that  in  theoretical  investigation  the 
induction  motor  is  best  treated  as  a  transformer,  and  the  elec- 
trical output  of  the  transformer  corresponds  to  the  mechanical 
output  of  the  induction  motor. 

The  secondary  of  the  motor  consists  of  two  or  more  circuits 
displaced  in  phase  from  each  other  so  as  to  offer  a  closed  sec- 
ondary to  the  primary  circuits,  irrespective  of  the  relative  motion. 
The  primary  consists  of  one  or  several  circuits. 

In  consequence  of  the  relative  motion  of  the  primary  and 
secondary,  the  magnetic  circuit  of  the  induction  motor  must  be 
arranged  so  that  the  secondary  while  revolving  does  not  leave  the 
magnetic  field  of  force.  That  means,  the  magnetic  field  of  force 
must  be  of  constant  intensity  in  all  directions,  or,  in  other  words, 
the  component  of  magnetic  flux  in  any  direction  in  space  be  of 
the  same  or  approximately  the  same  intensity  but  differing  in 
phase.  Such  a  magnetic  field  can  either  be  considered  as  the 
superposition  of  two  magnetic  fields  of  equal  intensity  in  quad- 
rature in  time  and  space,  or  it  can  be  represented  theoretically 
by  a  revolving  magnetic  flux  of  constant  intensity,  or  rotating 


INDUCTION  MACHINES  309 

field,  or  simply  treated  as  alternating  magnetic  flux  of  the  same 
intensity  in  every  direction. 

134.  The  operation  of  the  induction  motor  thus  can  also  be 
considered  as  due  to  the  action  of  a  rotating  magnetic  field  upon 
a  system  of  short-circuited  conductors.  In  the  motor  field  or 
primary,  usually  the  stator,  by  a  system  of  polyphase  impressed 
e.m.fs.  or  by  the  combination  of  a  single-phase  impressed  e.m.f. 
and  the  reaction  of  the  currents  produced  in  the  secondary,  a 
rotating  magnetic  field  is  produced.  This  rotating  field  produces 
currents  in  the  short-circuited  armature  or  secondary  winding, 
usually  the  rotor,  and  by  its  action  on  these  currents  drags  along 
the  secondary  conductors,  and  thus  speeds  up  the  armature  and 
tends  to  bring  it  up  to  synchronism,  that  is,  to  the  same  speed  as 
the  rotating  field,  at  which  speed  the  secondary  currents  would 
disappear  by  the  armature  conductors  moving  together  with  the 
rotating  field,  and  thus  cutting  no  lines  of  force.  The  secondary 
therefore  slips  in  speed  behind  the  speed  of  the  rotating  field  by 
as  much  as  is  required  to  produce  the  secondary  currents  and  give 
the  torque  necessary  to  carry  the  load.  The  slip  of  the  induction 
motor  thus  increases  with  increase  of  load,  and  is  approximately 
proportional  thereto.  Inversely,  if  the  secondary  is  driven  at  a 
higher  speed  than  that  of  the  rotating  field,  the  field  drags  the 
armature  conductors  back,  that  is,  consumes  mechanical  torque, 
and  the  machine  then  acts  as  a  brake  or  induction  generator. 

In  the  polyphase  induction  motor  this  magnetic  field  is  pro- 
duced by  a  number  of  electric  circuits  relatively  displaced  in 
space,  and  excited  by  currents  having  the  same  displacement  in 
phase  as  the  exciting  coils  have  in  space. 

In  the  single-phase  motor  one  of  the  two  superimposed  mag- 
netic quadrature  fields  is  excited  by  the  primary  electric  circuit, 
the  other  by  the .  secondary  currents  carried  into  quadrature 
position  by  the  rotation  of  the  secondary.  In  either  case,  at 
or  near  synchronism  the  magnetic  fields  are  practically  identical. 

The  transformer  feature  being  predominant,  in  theoretical 
investigations  of  induction  motors  it  is  generally  preferable  to 
start  therefrom. 

The  characteristics  of  the  transformer  are  independent  of  the 
ratio  of  transformation,  other  things  being  equal;  that  is,  dou- 
bling the  number  of  turns  for  instance,  and  at  the  same  time 
reducing  their  cross  section  to  one-half,  leaves  the  efficiency, 
regulation, etc.,  of  the  transformer  unchanged.  In  the  same  way, 


310     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

in  the  induction  motor  it  is  unessential  what  the  ratio  of  primary 
to  secondary  turns  is,  or,  in  other  words,  the  secondary  circuit 
can  be  wound  for  any  suitable  number  of  turns,  provided  the 
same  total  copper  cross  section  is  used.  In  consequence  hereof 
the  secondary  circuit  is  mostly  wound  with  one  or  two  bars  per 
slot,  to  get  maximum  amount  of  copper,  that  is,  minimum  resist- 
ance of  secondary. 

The  general  characteristics  of  the  induction  motor  being  inde- 
pendent of  the  ratio  of  turns,  it  is  for  theoretical  considera- 
tions simpler  to  assume  the  secondary  motor  circuits  reduced 
to  the  same  number  of  turns  and  phases  as  the  primary,  or  of  the 
ratio  of  transformation  1  to  1,  by  multiplying  all  secondary  cur- 
rents and  dividing  all  secondary  e.m.fs.  by  the  ratio  of  turns, 
multiplying  all  secondary  impedances  and  dividing  all  secondary 
admittances  by  the  square  of  the  ratio  of  turns,  etc. 

Thus  in  the  following  under  secondary  current,  e.m.f.,  impe- 
dance, etc.,  shall  always  be  understood  their  values  reduced  to 
the  primary,  or  corresponding  to  a  ratio  of  turns  1  to  1,  and  the 
same  number  of  secondary  as  primary  phases,  although  in  prac- 
tice a  ratio  1  to  1  will  hardly  ever  be  used,  as  not  fulfilling  the 
condition  of  uniform  effective  reluctance  desirable  in  the  start- 
ing of  the  induction  motor. 

II.  Polyphase  Induction  Motor 

1.  INTRODUCTION 

135.  The  typical  induction  motor  is  the  polyphase  motor. 
By  gradual  development  from  the  direct-current  shunt  motor 
we  arrive  at  the  polyphase  induction  motor. 

The  magnetic  field  of  any  induction  motor,  whether  supplied 
by  polyphase,  monocyclic,  or  single-phase  e.m.f.,  is  at  normal 
condition  of  operation,  that  is,  near  synchronism,  a  polyphase 
field.  Thus  to  a  certain  extent  all  induction  motors  can  be 
called  polyphase  machines.  When  supplied  with  a  polyphase 
system  of  e.m.fs.  the  internal  reactions  of  the  induction  motor 
are  simplest  and  only  those  of  a  transformer  with  moving  second- 
ary, while  in  the  single-phase  induction  motor  at  the  same  time 
a  phase  transformation  occurs,  the  second  or  magnetizing  phase 
being  produced  from  the  impressed  phase  of  e.m.f.  by  the  rota- 
tion of  the  motor,  which  carries  the  secondary  currents  into 
quadrature  position  with  the  primary  current. 


INDUCTION  MACHINES  311 

The  polyphase  induction  motor  of  the  three-phase  or  quarter- 
phase  type  is  the  one  most  commonly  used,  while  single-phase 
motors  have  found  a  more  limited  application  only,  and  especially 
for  smaller  powers. 

Thus  in  the  following  more  particularly  the  polyphase  induc- 
tion machine  shall  be  treated,  and  the  single-phase  type  discussed 
only  in  so  far  as  it  differs  from  the  typical  polyphase  machine. 

2.  CALCULATION 

136.  In  the  polyphase  induction  motor, 
Let 

Y  =  g  —  jb  =  primary  exciting  admittance,  or  admit- 
tance of  the  primary  circuit  with  open  secondary 
circuit; 
that  is, 

ge  =  magnetic  power  current, 

be  =  wattless  magnetizing  current, 

where  e  =  counter-generated  e.m.f.  of  the  motor; 

ZQ  =  r0  +  jxQ  =  primary  self -inductive  impedance,  and 
Zi  =  7*1  +  jxi  =  secondary    self-inductive    impedance, 
reduced  to  the  primary  by  the  ratio  of  turns.1 

All  these  quantities  refer  to  one  primary  circuit  and  one  corre- 
sponding secondary  circuit.  Thus  in  a  three-phase  induction 
motor  the  total  power,  etc.,  is  three  times  that  of  one  circuit,  in 
the  quarter-phase  motor  with  three-phase  armature  1J^  of  the 
three  secondary  circuits  are  to  be  considered  as  corresponding 
to  each  of  the  two  primary  circuits,  etc. 

Let  e  =  primary  counter-generated  e.m.f.,  or  e.m.f.  generated 
in  the  primary  circuit  by  the  flux  interlinked  with  primary  and 
secondary  (mutual  induction);  s  =  slip,  with  the  primary  fre- 
quency as  unit;  that  is,  s  =  0  denoting  synchronous  rotation, 
s  =  l  standstill  of  the  motor. 

We  then  have 

1  —  s  =  speed  of  the  motor  secondary  as  fraction  of  syn- 
chronous speed, 

sf  =  frequency  of  the  secondary  currents, 
where 

/  =  frequency  impressed  upon  the  primary; 

1  The  self -inductive  reactance  refers  to  that  flux  which  surrounds  one  of 
the  electric  circuits  only,  without  being  interlinked  with  the  other  circuits. 


312     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

hence,   . 

se  =  e.m.f.  generated  in  the  secondary. 

The  actual  impedance  of  the  secondary  circuit  at  the  frequency 
sf  is 

Zi8  =  7*1  +jsxi; 

hence,  the  secondary  current  is 
se  se 


where 


the  primary  exciting  current  is 

/oo  =eY  =  e[g  —  jb], 
and  the  total  primary  current  is 

/o  =  e  I  (ai  -f  g)  —  j  (a2  +  b] 
where 

The  e.m.f.  consumed  in  the  primary  circuit  by  the  impedance 
ZQ  is  /oZo,  the  counter-generated  e.m.f.  is  e,  hence,  the  primary 
terminal  voltage  is 

EQ  =  e  +  IQZQ  =  e[l  +  (bi  —  j&2)  (r0  +  jx0)]  .=  e  (ci  —  jc2), 

where 

Ci  =  1  +  robi  +  Xobz  and  c2  =  r062  —  Xobi. 

Eliminating  complex  quantities,  we  have 

EQ  =  e  Vci2  +  c22, 
hence,  the  counter-generated  e.m.f.  of  motor, 

e  =  — ==  , 

where 

EQ  =  impressed  e.m.f.,  absolute  value. 

Substituting  this  value  in  the  equations  of  /i,  /oo,  /o,  etc  , 
gives  the  complex  expressions  of  currents  and  e.m.fs.,  and  elimi- 
nating the  imaginary  quantities  we  have  the  primary  current, 

/o  =  e  V&i2  +  622,  etc. 


INDUCTION  MACHINES  313 

The  torque  of  the  polyphase  induction  motor  (or  any  other 
motor  or  generator)  is  proportional  to  the  product  of  the  mutual 
magnetic  flux  and  the  component  of  ampere-turns  of  the  sec- 
ondary, which  is  in  phase  with  the  magnetic  flux  in  time,  but  in 
quadrature  therewith  in  direction  or  space.  Since  the  generated 
e.m.f.  is  proportional  to  the  mutual  magnetic  flux  and  the  num- 
ber of  turns,  but  in  quadrature  thereto  in  time,  the  torque  of  the 
induction  motor  is  proportional  also  to  the  product  of  the  gen- 
erated e.m.f.  and  the  component  of  secondary  current  in  quadra- 
ture therewith  in  time  and  in  space. 

Since  /i  =  e  (a\  —  ja2)  is  the  secondary  current  corresponding 

to  the  generated  e.m.f.  e,  the  secondary  current  in  the  quadrature 
position  thereto  in  space,  that  is,  corresponding  to  the  e.m.f.  je,  is 

jli  =  e(a2 


and  die  is  the  component  of  this  current  in  quadrature  in  time 
with  the  e.m.f.  e. 

Thus  the  torque  is  proportional  toe  X  die,  or 

D  =  ezdi 


n2  +  s*xi*  '    (ex2  +  c22)  (n2  +  sV) 

This  value  D  is  in  its  dimension  a  power,  and  it  is  the  power 
which  the  torque  of  the  motor  would  develop  at  synchronous 
speed. 

137.  In  induction  motors,  and  in  general  motors  which  have 
a  definite  limiting  speed,  it  is  preferable  to  give  the  torque  in  the 
form  of  the  power  developed  at  the  limiting  speed,  in  this  case 
synchronism,  as  "synchronous  watts,"  since  thereby  it  is  made 
independent  of  the  individual  conditions  of  the  motor,  as  its 
number  of  poles,  frequency,  etc.,  and  made  comparable  with  the 
power  input,  etc.  It  is  obvious  that  when  given  in  synchronous 
watts,  the  maximum  possible  value  of  torque  which  could  be 
reached,  if  there  were  no  losses  in  the  motor,  equals  the  power 
input.  Thus,  in  an  induction  motor  with  9000  watts  power 
input,  a  torque  of  7000  synchronous  watts  means  that  %  of  the 
maximum  theoretically  possible  torque  is  realized,  while  the 
statement,  "a  torque  of  30  pounds  at  1-foot  radius,"  would  be 
meaningless  without  knowing  the  number  of  poles  and  the  fre- 
quency. Thus,  the  denotation  of  the  torque  in  synchronous 


314     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

watts  is   the   most   general,  and  preferably  used  in  induction 
motors. 

Since  the  theoretical  maximum  possible  torque  equals  the 
power  input,  the  ratio 

torque  in  synchronous  watts  output 

power  input 
that  is, 

actual  torque 

maximum  possible  torque* 

is  called  the  torque  efficiency  of  the  motor,  analogous  to  the  power 
efficiency  or 

power  output  t 
power  input 
that  is, 

power  output 

maximum  possible  power  output  * 
Analogously 

torque  in  synchronous  watts 
volt-amperes  input 

is  called  the  apparent  torque  efficiency. 

The  definitions  of  these  quantities,  which  are  of  importance  in 
judging  induction  motors,  are  thus: 

The  "efficiency"  or  "power  efficiency"  is  the  ratio  of  the  true 
mechanical  output  of  the  motor  to  the  output  which  it  would 
give  at  the  same  power  input  if  there  were  no  internal  losses  in 
the  motor. 

The  "apparent  efficiency"  or  "apparent  power  efficiency"  is  the 
ratio  of  the  mechanical  output  of  the  motor  to  the  output  which 
it  would  give  at  the  same  volt-ampere  input  if  there  were  neither 
internal  losses  nor  phase  displacement  in  the  motor. 

The  "torque  efficiency"  is  the  ratio  of  the  torque  of. the  motor 
to  the  torque  which  it  would  give  at  the  same  power  input  if 
there  were  no  internal  losses  in  the  motor. 

The  "apparent  torque  efficiency"  is  the  ratio  of  the  torque  of 
the  motor  to  the  torque  which  it  would  give  at  the  same  volt- 
ampere  input  if  there  were  neither  internal  losses  nor  phase  dis- 
placement in  the  motor. 

The  torque  efficiencies  are  of  special  interest  in  starting  where 
the  power  efficiencies  are  necessarily  zero,  but  it  nevertheless 


INDUCTION  MACHINES  315 

is  of  importance  to  find  how  much  torque  per  watt  or  per  volt- 
ampere  input  is  given  by  the  motor. 

Since  D  =  e2ai  is  the  power  developed  by  the  motor  torque 
at  synchronism,  the  power  developed  at  the  speed  of  (1  —  s) 
X  synchronism,  or  the  actual  power  output  of  the  motor,  is 

P  =  (1  -  s)  D 
=  e2ai  (1  -  s) 
eViS  (1  -  s) 


The  output  P  includes  friction,  windage,  etc.  ;  thus,  the  net  me- 
chanical output  is  P  —  friction,  etc.  Since,  however,  friction, 
etc.,  depend  upon  the  mechanical  construction  of  the  individual 
motor  and  its  use,  it  cannot  be  included  in  a  general  formula. 
P  is  thus  the  mechanical  output,  and  D  the  torque  developed  at 
the  armature  conductors. 

The  primary  current 

IQ  =  e  (bi  —  jb2) 

has  the  quadrature  components  ebi  and  ebz. 
The  primary  impressed  e.m.f. 

EQ  =  e  (ci  -  jc2) 

has  the  quadrature  components  eci  and  ec2. 

Since  the  components  ebi  and  ec2,  and  eb2  and  eci,  respectively, 
are  in  quadrature  with  each  other,  and  thus  represent  no  power, 
the  power  input  of  the  primary  circuit  is 

ebl  X  eci  +  eb2  X  ec2, 
or  P0  =  e2  (bid  +  62c2). 

The  volt-amperes  or  apparent  input  is  obviously, 
Pa 


+  &22)  (d2  +c22). 

138.  These  equations  can  be  greatly  simplified  by  neglecting 
the  exciting  current  of  the  motors,  and  approximate  values  of 
current,  torque,  power,  etc.,  derived  thereby,  which  are  suffi- 
ciently accurate  for  preliminary  investigations  of  the  motor  at 
speeds  sufficiently  below  synchronism  to  make  the  total  motor 
current  large  compared  with  the  exciting  current. 


316     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

In  this  case  the  primary  current  equals  the  secondary  current, 
that  is, 


T  se 

IQ  =  /i  ==  --  =  e  (oi  -  ja2), 


where 
and 


=  <M(l  +  lr7    =£L-4r; 


=   6 


and,  in  absolute  values, 


e 


hence, 


and  the  torque,  in  synchronous  watts,  is 
D  =  ezai  =  - 

TV  -f 

hence,  substituting  for  e, 


D  = 


-f  sr0)2  +s2  (a?!  + 
and  the  power  is 

s  (1  —  s)  eQ2r} 


P  =  — 

If  the  additional  resistance  r  is  inserted  into  the  armature 
circuit,  and  the  total  armature  resistance  thus  becomes  7*1  +  r, 
instead  of  r\t  substituting  (r\  +  r)  in  above  equations  we  have 

seQ2  (n  +  r) 
and 

^    =   7 JL JL \2°    i        a/ JL~~\2»   6^C* 

Neglecting  also  the  primary  self-inductive  impedance,  ZQ  = 


INDUCTION  MACHINES  317 

7*0  H-  J£o,  which  sometimes  can  be  done  as  first  approximation, 
especially  at  large  values  of  r,  these  equations  become 

D  seo2  (n  +  r) 

P  =  s  —, —  ,    12    2   >  etc. 

Oi+r)2  +  s2zi2 

139.  Since  the  counter-generated  e.m.f.  e  (and  thus  the  im- 
pressed e.m.f.  EQ)  enters  in  the  equation  of  current,  magnetism, 
etc.,  as  a  simple  factor,  in  the  equations  of  torque,  power  input 
and  output,  and  volt-ampere  input  as  square,  and  cancels  in  the 
equation  of  efficiency,   power-factor,  etc.,  it  follows  that  the 
current,  magnetic  flux,  etc.,  of  an  induction  motor  are  propor- 
tional to  the  impressed  e.m.f.,  the  torque,  power  output,  power 
input,  and  volt-ampere  input  are  proportional  to  the  square  of 
the  impressed  e.m.f.,  and  the  torque-  and  power  efficiencies  and 
the  power-factor  are  independent  of  the  impressed  voltage. 

In  reality,  however,  a  slight  decrease  of  efficiency  and  power- 
factor  occurs  at  higher  impressed  voltages,  due  to  the  increase 
of  resistance  caused  by  the  increasing  temperature  of  the  motor 
and  due  to  the  approach  to  magnetic  saturation,  and  a  slight 
decrease  of  efficiency  occurs  at  lower  voltages  when  including 
in  the  efficiency  the  loss  of  power  by  friction,  since  this  is  inde- 
pendent of  the  output  and  thus  at  lower  voltage,  that  is,  lesser 
output,  a  larger  percentage  of  the  output,  so  that  the  efficiencies 
and  the  power-factor  can  be  considered  as  independent  of  the 
impressed  voltage,  and  the  torque  and  power  proportional  to  the 
square  thereof  only  approximately,  but  sufficiently  close  for  many 
purposes. 

3.  LOAD  AND  SPEED  CURVES 

140.  The  calculation  of  the  induction  motor  characteristics 
is  most  conveniently  carried  out  in  tabulated  form  by  means 
of  above-given  equations  as  follows: 

Let  ZQ  =  r0  +  JXQ  =  0.1  -f-  0.3  j  =  primary  self-inductive  im- 
pedance. 

Zi  =  TI  +  jxi  =  0.1  +  0.3  j  =  secondary     self-inductive 

impedance  reduced  to  pri- 
mary. 

Y  =  g  —  jb  =  0.01  —  0.1  j  =  primary    exciting    admit- 
tance. 
EQ  =  110  volts  =  primary  impressed  e.m.f. 


318     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


It  is  then,  per  phase, 


„ 

1 

rO 

«„ 

%, 

X 

n% 

[V* 

H| 

s 

'  + 

-0 

*f 

it 

§ 

1  ! 

• 
\ 

h 
> 

»*r« 

:? 

i* 

0 

0.0100 

0 

3 

0.010.10 

1.031 

+0.007 

1.031106.6 

0.101010.8 

0.01 

o.oioolo.ioo 

3.003 

0.11 

0.103 

1.042 

-0.0231.042 

105.7 

0.1507 

15.9 

0.02 

0.0100 

0.200 

3.012 

0.21 

0.112 

1.055 

-0.052 

1.056 

104.3 

0.238 

24.8 

0.05 

0.0102 

0.490 

3.073 

0.50 

0.173 

1.102 

-0.133 

1.110 

99.20.522 

51.8 

0.1 

0.0109 

0.920 

3.276 

0.93 

0.376 

1.206 

-0.241 

1.230 

89.5 

1.003 

89.7 

0.15 
0.2 

0.0120 
0.0136 

1.25 
1.47 

3.563 
3.883 

1.26 
1.48 

0.663 
0.983 

1.325 
1.443 

-0.308 
-0.354 

1.360 
1.485 

80.91.424 

74.21.777 

115 
132 

0.3 
0.5 
1.0 

0.01811.66 
0.03251.54 
O.lOOOjl.OO 

L.49 
2.31 
5.00 

1.671.50 
1.552.41 
1.013.10 

1.617 

1.878 
2.031 

-0.351 
-0.224 
+0.007 

1.654 
1.891 
2.031 

66.6 

58.2 
54.1 

2.245 
2.865 
3.261 

149 
167 
176 

s 

e* 

D  = 
e2oi 

P  = 

Pa  = 

Eol 

p  = 

bici  + 

&2C2 

Po2  = 

eff.  = 
P 
Po 

app.  eff.  = 
P 

Pa 

pow.fac.  = 
Po 

Pa 

0 

11,360 

0 

0 

1.19 

0.011 

0.125 

0 

0 

10.5 

0.01 

11,170 

1.117 

1. 

106 

1.75 

0.112 

1.249 

88.5 

63.2 

71.5 

0.02 

10,880 

2.17C 

.  2. 

133 

2.73 

0.216 

2.350 

91.0 

78.3 

86.2 

0.05 

9,840 

4.82 

4. 

58 

5.70 

0.528 

5.20 

88.3 

80.5 

91.3 

0.1 

8,010 

7.38 

6. 

64 

9.87 

1.030 

8.25 

80.7 

67.3 

83.5 

0.15 

6,540 

8.20 

6. 

97 

12.65 

1.466 

9.60 

72.5 

55.0 

76.0 

0.2 

5,510 

8.10 

6. 

48 

14.52 

1.782 

9.80 

66.0 

44.6 

67.5 

0.3 

4,440 

7.36 

5. 

15 

16.4 

2.154 

9.55 

53.8 

31.5 

58.3 

0.5 

3,390 

5.23 

2. 

61 

18.4 

2.370 

8.04 

32.3 

14.2 

43.8 

1.0 

2,930 

2.93 

0 

19.4 

2.072 

6.08 

0 

0 

31.3 

Diagrammatically  it  is  most  instructive  in  judging  about  an 
induction  motor  to  plot  from  the  preceding  calculation — 

1st.  The  load  curves,  that  is,  with  the  load  or  power  output  as 
abscissas,  the  values  of  speed  (as  a  fraction  of  synchronism),  of 
current  input,  power-factor,  efficiency,  apparent  efficiency,  and 
torque. 

2d.  The  speed  curves,  that  is,  with  the  speed,  as  a  fraction  of 
synchronism,  as  abscissas,  the  values  of  torque,  current  input, 
power-factor,  torque  efficiency,  and  apparent  torque  efficiency. 

The  load  curves  are  most  instructive  for  the  range  of  speed 
near  synchronism,  that  is,  the  normal  operating  conditions  of 
the  motor,  while  the  speed  curves  characterize  the  behavior  of 
the  motor  at  any  speed. 


INDUCTION  MACHINES 


319 


In  Fig.  176  are  plotted  the  load  curves,  and  in  Fig  177  the 
speed  curves  of  a  typical  polyphase  induction  motor  of  moderate 
size,  having  the  folio  wing  constants:  eQ  =  110;  Y  =  0.01  —  0.1  j; 
Z,  =  0.1  +  0.3  j,  and  Z0  =  0.1  +  0.3  j. 

As  sample  of  a  poor  motor  of  high  resistance  and  high  admit- 
tance or  exciting  current  are  plotted  in  Fig.  178  the  load  curves 
of  a  motor  having  the  following  constants:  eQ  =  110;  Y  =  0.04 


Z0=Zf=0. 1+0.3  j     .Y— 0.01 -0.1  j 


2000  3000  4000  5000 

FIG.  176. — Induction  motor  load 


cooo 
curves. 


-  0.4  j',  Zi  =  0.3  +  0.3  j,  and  ZQ  =  0.3  +  0.3  j,  showing  the 
overturn  of  the  power-factor  curve  frequently  met  in  poor  motors. 

141.  The  shape  of  the  characteristic  motor  curves  depends 
entirely  on  the  three  complex  constants,  Y,  Zi,  and  ZQ,  but  is 
essentially  independent  of  the  impressed  voltage. 

Thus  a  change  of  the  admittance  Y  has  no  effect  on  the  char- 
acteristic curves,  provided  that  the  impedances  Z\  and  Z0  are 


320     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

changed  inversely  proportional  thereto,  such  a  change  merely 
representing  the  effect  of  a  change  of  impressed  voltage.     A 


1.0  0.9  0.8  0.7  0.6  0.5  0.4  0.3  0.2  0.1 

FIG.  177. — Induction  motor  speed  curves. 


1000  2000  3000  4000 

FIG.  178. — Load  curves  of  poor  induction  motor. 

moderate  change  of  one  of  the  impedances  has  relatively  little 
effect  on  the  motor  characteristics,  provided  that  the  other 
impedance  changes  so  that  the  sum  Zi  +  ZQ  remains  constant, 


INDUCTION  MACHINES  321 

and  thus  the  motor  can  be  characterized  by  its  total  internal 
impedance,  that  is, 

Z  =  Zl  +  Z0; 
or  r  +  jx  =  (ri  +  r0)  +  j  (xi  +  XQ). 

Thus  the  characteristic  behavior  of  the  induction  motor  de- 
pends upon  two  complex  imaginary  constants,  Y  and  Z,  or  four 
real  constants,  g,  6,  r,  x,  the  same  terms  which  characterize  the 
stationary  alternating-current  transformer  on  non-inductive  load. 

Instead  of  conductance  g,  susceptance  6,  resistance  r,  and  react- 
ance x,  as  characteristic  constants  may  be  chosen:  the  absolute 
exciting  admittance  y  =  \/g2  -f-  &2;  the  absolute  self-inductive 
impedance  z  —  \/r2-}-x2',  the  power-factor  of  admittance  0  = 
g/y,  and  the  power-factor  of  impedance  a  =  r/z. 

142.  If  the  admittance  y  is  reduced  rz-fold  and  the  impedance 
z  increased  n-fold,  with  the  e.m.f.  \SnEQ  impressed  upon  the 
motor,  the  speed,  torque,  power  input  and  output,  volt-ampere 
input  and  excitation,  power-factor,  efficiencies,  etc.,  of  the  motor, 
that  is,  all  its  characteristic  features,  remain  the  same,  as  seen 
from  above  given  equations,  and  since  a  change  of  impressed 
e.m.f.  does  not  change  the  characteristics,  it  follows  that  a  change 
of  admittance  and  of  impedance  does  not  change  the  character- 
istics of  the  motor  provided  the  product  7  =  yz  remains  the  same. 

Thus  the  induction  motor  is  characterized  by  three  constants 
only: 

The  product  of  exciting  admittance  and  self-inductive  impe- 
dance 7  =  yz,  which  may  be  called  the  characteristic  constant 
of  the  motor. 

The  power-factor  of  exciting  admittance  /?  =  -• 

y 

The  power-factor  of  self-inductive  impedance  a  =  -- 

All  these  three  quantities  are  absolute  numbers. 

The  physical  meaning  of  the  characteristic  constant  or  the  prod- 
uct of  the  exciting  admittance  and  impedance  is  the  following: 

If  IQQ  =  exciting  current  and  7i0  =  starting  current,  we  have, 
approximately, 


E0 
z  =  j-i 

•MO 

Jf  00 
y    =    yZ    =         ~  • 


322     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

The  characteristic  constant  of  the  induction  motor  7  =  yz  is 
the  ratio  of  exciting  current  to  starting  current  or  current  at 
standstill. 

At  given  impressed  e.m.f.,  the  exciting  current  7oo  is  inversely 
proportional  to  the  mutual  inductance  of  primary  and  secondary 
circuit.  The  starting  current  Iio  is  inversely  proportional  to 
the  sum  of  the  self-inductance  of  primary  and  secondary  circuit. 

Thus  the  characteristic  constant  7  =  yz  is  approximately  the 
ratio  of  total  self-inductance  to  mutual  inductance  of  the  motor 
circuits;  that  is,  the  ratio  of  the  flux  interlinked  with  only  one 
circuit,  primary  or  secondary,  to  the  flux  interlinked  with  both 
circuits,  primary  and  secondary,  or  the  ratio  of  the  waste  or 
leakage  flux  to  the  useful  flux.  The  importance  of  this  quantity 
is  evident. 

4.  EFFECT  OF  ARMATURE  RESISTANCE  AND  STARTING 

143.  The  secondary  or  armature  resistance  TI  enters  the  equa- 
tion of  secondary  current  thus: 


S6  I         STi  .          S*Xi 


-3 


\ 

)  • 


and  the  further  equations  only  indirectly  in  so  far  as  TI  is  con- 
tained in  ai  and  a2. 

Increasing  the  armature  resistance  n-fold,  to  nri,  we  get  at  an 
n-fold  increased  slip  ns, 

use  se 

1  " 


n  +  jsx 

that  is,  the  same  value,  and  thus  the  same  values  for  e,  Jo,  D, 
Po,  Pa,  while  the  power  is  decreased  from  P  =  (1  —  s)  D  to 
P  =  (1  —  ns)  D,  and  the  efficiency  and  apparent  efficiency  are 
correspondingly  reduced.  The  power-factor  is  not  changed; 
hence,  an  increase  of  armature  resistance  ri  produces  a  propor- 
tional increase  of  slip  s,  and  thereby  corresponding  decrease  of 
power  output,  efficiency  and  apparent  efficiency,  but  does  not 
change  the  torque,  power  imput,  current,  power-factor,  and  the 
torque  efficiencies. 

Thus  the  insertion  of  resistance  in  the  armature  or  secondary 
of  the  induction  motor  offers  a  means  of  reducing  the  speed 
corresponding  to  a  given  torque,  and  thereby  the  desired  torque 
can  be  produced  at  any  speed  below  that  corresponding  to  short- 


INDUCTION  MACHINES 


323 


circuited  armature  or  secondary  without  changing  the  input  or 
current. 

Hence,  given  the  speed  curve  of  a  short-circuited  motor,  the 
speed  curve  with  resistance  inserted  in  the  armature  can  be 
derived  therefrom  directly  by  increasing  the  slip  in  proportion 
to  the  increased  resistance. 


1.0     0.9       08 


SLIP  FRACTION 


SLIP  FRACTION    C  F  8YNC 


00       05        04       03       02       0.1 


Y  »  0.01  -  0. 


X 


HO 


TORQU • 
WATT« 

SS& 


2000 


1.0      0.9       0.8       0.7       0.0       0.5        0.4       0.3        0.2       0.1        0 

FIG.  179. — Induction  motor  speed-torque  and  -current  curves. 


This  is  done  in  Fig.  179,  in  which  are  shown  the  speed  curves 
of  the  motor  Figs.  176  and  177,  between  standstill  and  syn- 
chronism, for — 

Short-circuited  armature,  n  =  0.1  (same  as  Fig.  177). 

0.15  ohm  additional  resistance  per  circuit  inserted  in  armature, 
r\  =  0.25,  that  is,  2.5  times  increased  slip. 


324     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

0.5  ohm  additional  resistance  inserted  in  the  armature, 
r\  =  0.6,  that  is,  6  times  increased  slip. 

1.5  ohm  'additional  resistance  inserted  in  the  armature, 
r\  =  1.6,  that  is,  16  times  increased  slip. 

The  corresponding  current  curves  are  shown  on  the  same  sheet. 

With  short-circuited  secondary  the  maximum  torque  of  8250 
synchronous  watts  is  reached  at  16  per  cent.  slip.  The  starting 
torque  is  2950  synchronous  watts,  and  the  starting  current  176 
amp. 

With  armature  resistance  TI  =  0.25,  the  same  maximum 
torque  is  reached  at  40  per  cent,  slip,  the  starting  torque  is  in- 
creased to  6050  synchronous  watts,  and  the  starting  current 
decreased  to  160  amp. 

With  the  secondary  resistance  r\  =  0.6,  the  maximum  torque 
of  8250  synchronous  watts  approximately  takes  place  in  start- 
ing, and  the  starting  current  is  decreased  to  124  amp. 

With  armature  resistance  7*1  =  1.6,  the  starting  torque  is 
below  the  maximum,  5620  synchronous  watts,  and  the  starting 
current  is  only  64  amp. 

In  the  two  latter  cases  the  lower  or  unstable  branch  of  the 
torque  curve  has  altogether  disappeared,  and  the  motor  speed 
is  stable  over  the  whole  range;  the  motor  starts  with  the  maxi- 
mum torque  which  it  can  reach,  and  with  increasing  speed, 
torque  and  current  decrease;  that  is,  the  motor  has  the  character- 
istic of  the  direct-current  series  motor,  except  that  its  maximum 
speed  is  limited  by  synchronism. 

144.  It  follows  herefrom  that  high  secondary  resistance,  while 
very  objectionable  in  running  near  synchronism,  is  advantageous 
in  starting  or  running  at  very  low  speed,  by  reducing  the  current 
input  and  increasing  the  torque. 

In  starting  we  have 

s  =  1. 

Substituting  this  value  in  the  equations  of  subsection  2  gives 
the  starting  torque,  starting  current,  etc.,  of  the  polyphase  in- 
duction motor. 

In  Fig.  180  are  shown  for  the  motor  in  Figs.  176,  177  and  179 
the  values  of  starting  torque,  current,  power-factor,  torque 
efficiency,  and  apparent  torque  efficiency  for  various  values  of 
the  secondary  motor  resistance,  from  r\  =  0.1,  the  internal  re- 
sistance of  the  motor,  or  R  =  0  additional  resistance  to  n  =  5.1 


INDUCTION  MACHINES 


325 


or  R  =  5  ohms  additional  resistance.  The  best  values  of  torque 
efficiency  are  found  beyond  the  maximum  torque  point. 

The  same  Fig.  180  also  shows  the  torque  with  resistance  in- 
serted into  the  primary  circuit. 

The  insertion  of  reactance,  either  in  the  primary  or  in  the 
secondary,  is  just  as  unsatisfactory  as  the  insertion  of  resistance 
in  the  primary  circuit. 


1.0        1.5       2.0        2.5       3.0       3.5       4.0       4.5       5.0 


FIG.  180. — Induction    motor    starting    torque    with    resistance 

secondary. 


in    the 


Capacity  inserted  in  the  secondary  very  greatly  increases  the 
torque  within  the  narrow  range  of  capacity  corresponding  to 
resonance  with  the  internal  reactance  of  the  motor,  and  the 
torque  which  can  be  produced  in  this  way  is  far  in  excess  of  the 
maximum  torque  of  the  motor  when  running  or  when  starting 
with  resistance  in  the  secondary. 


326     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

But  even  at  its  best  value,  the  torque  efficiency  available 
with  capacity  in  the  secondary  is  below  that  available  with 
resistance. 

For  further  discussion  of  the  polyphase  inductance  motor,  see 
"Theory  and  Calculation  of  Alternating-current  Phenomena." 

in.  Single -phase  Induction  Motor 

•1.  INTRODUCTION 

146.  In  the  polyphase  motor  a  number  of  secondary  coils 
displaced  in  position  from  each  other  are  acted  upon  by  a  num- 
ber of  primary  coils  displaced  in  position  and  excited  by  e.m.fs. 
displaced  in  phase  from  each  other  by  the  same  angle  as  the  dis- 
placement of  position  of  the  coils. 

In  the  single-phase  induction  motor  a  system  of  secondary 
circuits  is  acted  upon  by  one  primary  coil  (or  system  of  primary 
coils  connected  in  series  or  in  parallel)  excited  by  a  single  alter- 
nating current. 

A  number  of  secondary  circuits  displaced  in  position  must  be 
used  so  as  to  offer  to  the  primary  circuit  a  short-circuited  sec- 
ondary in  any  position  of  the  armature.  If  only  one  secondary 
coil  is  used,  the  motor  is  a  synchronous  induction  motor  and 
belongs  to  the  class  of  reaction  machines. 

A  single-phase  induction  motor  will  not  start  from  rest,  but 
when  started  in  either  direction  will  accelerate  with  increasing 
torque  and  approach  synchronism. 

When  running  at  or  very  near  synchronism,  the  magnetic  field 
of  the  single-phase  induction  motor  is  practically  identical  with 
that  of  a  polyphase  motor,  that  is,  can  be  represented  by  the 
theory  of  the  rotating  field.  Thus,  in  a  turn  wound  under  angle 
r  to  the  primary  winding  of  the  single-phase  induction  motor,  at 
synchronism  an  e.m.f.  is  generated  equal  to  that  generated  in  a 
turn  of  the  primary  winding,  but  differing  therefrom  by  angle 
6  =  T  in  time  phase. 

In  a  polyphase  motor  the  magnetic  flux  in  any  direction  is  due 
to  the  resultant  m.m.f.  of  primary  and  of  secondary  currents,  in 
the  same  way  as  in  a  transformer.  The  same  is  the  case  in  the 
direction  of  the  axis  of  the  exciting  coil  of  the  single-phase  induc- 
tion motor.  In  the  direction  at  right  angles  to  the  axis  of  the 
exciting  coil,  however,  the  magnetic  flux  is  due  to  the  m.m.f.  of 


INDUCTION  MACHINES  327 

the  secondary  currents  alone,  no  primary  e.m.f.  acting  in  this 
direction. 

Consequently,  in  the  polyphase  motor  running  synchronously, 
that  is,  doing  no  work  whatever,  the  secondary  becomes  current- 
less,  and  the  primary  current  is  the  exciting  current  of  the  motor 
only.  In  the  single-phase  induction  motor,  even  when  running 
light,  the  secondary  still  carries  the  exciting  current  of  the  mag- 
netic flux  in  quadrature  with  the  axis  of  the  primary  exciting 
coil.  Since,  this  flux  has  essentially  the  same  intensity  as  the 
flux  in  the  direction  of  the  axis  of  the  primary  exciting  coil,  the 
current  in  the  armature  of  the  single-phase  induction  motor  run- 
ning light,  and  therefore  also  the  primary  current  corresponding 
thereto,  has  the  same  m.m.f.,  that  is,  the  same  intensity,  as 
the  primary  exciting  current,  and  the  total  primary  current  of 
the  single-phase  induction  motor  running  light  is  thus  twice  the 
exciting  current,  that  is,  it  is  the  exciting  current  of  the  main 
magnetic  flux  plus  the  current  producing  in  the  secondary  the 
exciting  current  of  the  cross  magnetic  flux.  In  reality  it  is 
slightly  less,  especially  in  small  motors,  due  to  the  drop  of  voltage 
in  the  self-inductive  impedance  and  the  drop  of  quadrature  mag- 
netic flux  below  the  impressed  primary  magnetic  flux  caused 
thereby.  In  the  secondary  at  synchronism  this  secondary 
exciting  current  is  a  current  of  twice  the  primary  frequency;  at 
any  other  speed  it  is  of  a  frequency  equal  to  speed  (in  cycles)  plus 
synchronism. 

Thus,  if  in  a  quarter-phase  motor  running  light  one  phase  is 
open-circuited,  the  current  in  the  other  phase  doubles.  If  in  the 
three-phase  motor  two  phases  are  open-circuited,  the  current  in 
the  third  phase  trebles,  since  the  resultant  m.m.f.  of  a  three- 
phase  machine  is  1.5  times  that  of  one  phase.  In  consequence 
thereof,  the  total  volt-ampere  input  of  the  motor  remains  the 
same  and  at  the  same  magnetic  density,  or  the  same  impressed 
e.m.f.,  all  induction  motors,  single-phase  as  well  as  polyphase, 
consume  approximately  the  same  volt-ampere  input,  and  the 
same  power  input  for  excitation,  and  give  the  same  distribution 
of  magnetic  flux. 

146.  Since  the  maximum .  output  of  a  single-phase  motor  at 
the  same  impressed  e.m.f.  is  considerably  less  than  that  of  a  poly- 
phase motor,  it  follows  therefrom  that  the  relative  exciting  cur- 
rent in  the  single-phase  motor  must  be  larger. 

The  cause  of  this  cross  magnetization  in  the  single-phase  indue- 


328     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

tion  motor  near  synchronism  is  that  the  secondary  armature 
currents  lag  90  deg.  behind  the  magnetism,  and  are  carried  by 
the  synchronous  rotation  90  deg.  in  space  before  reaching  their 
maximum,  thus  giving  the  same  magnetic  effect  as  a  quarter- 
phase  e.m.f.  impressed  upon  the  primary  system  in  quadrature 
position  with  the  main  coil.  Hence  they  can  be  eliminated  by 
impressing  a  magnetizing  quadrature  e.m.f.  upon  an  auxiliary 
motor  circuit,  as  is  done  in  the  monocyclic  motor. 

Below  synchronism,  the  secondary  currents  are  carried  less 
than  90  deg.,  and  thus  the  cross  magnetization  due  to  them  is 
correspondingly  reduced,  and  becomes  zero  at  standstill. 

The  torque  is  proportional  to  the  power  component  of  the 
armature  currents  times  the  intensity  of  magnetic  flux  in  quad- 
rature position  thereto. 

In  the  single-phase  induction  motor,  the  armature  power 
currents  I'\  =  ea\  can  exist  only  coaxially  with  the  primary 
coil,  since  this  is  the  only  position  in  which  corresponding  pri- 
mary currents  can  exist.  The  magnetic  flux  in  quadrature  posi- 
tion is  proportional  to  the  component  of  e  carried  in  quadrature, 
or  approximately  to  (1  —  s)  e,  and  the  torque  is  thus 

D  =  (1  -  s)  el'  =  (1  -  s)  e2alf 

thus  decreases  much  faster  with  decreasing  speed,  and  becomes 
zero  at  standstill.  The  power  is  then 

P  =  (1  -  sYel'  =  (1  -  s)262a!. 

Since  in  the  single-phase  motor  only  one  primary  circuit  but 
a  multiplicity  of  secondary  circuits  exist,  all  secondary  circuits 
are  to  be  considered  as  corresponding  to  the  same  primary  cir- 
cuit, and  thus  the  joint  impedance  of  all  secondary  circuits  must 
be  used  as  the  secondary  impedance,  at  least  at  or  near  syn- 
chronism. Thus,  if  the  armature  has  a  quarter-phase  winding 
of  impedance  Zi  per  circuit,  the  resultant  secondary  impedance  is 

r? 

sr;  if  it  contains  a  three-phase  winding  of  impedance  Zi  per 

a 

17 

circuit,  the  resultant  secondary  impedance  is  •=- 

In  consequence  hereof  the  resultant  secondary  impedance  of  a 
single-phase  motor  is  less  in  comparison  with  the  primary  im- 
pedance than  in  the  polyphase  motor.  Since  the  drop  of  speed 
under  load  depends  upon  the  secondary  resistance,  in  the  single- 


INDUCTION  MACHINES  329 

phase  induction  motor  the  drop  in  speed  at  load  is  generally  less 
than  in  the  polyphase  motor;  that  is,  the  single-phase  induction 
motor  has  a  greater  constancy  of  speed  than  the  polyphase 
induction  motor,  but  just  as  the  polyphase  induction  motor,  it 
can  never  reach  complete  synchronism,  but  slips  below  synchro- 
nism, approximately  in  proportion  to  the  speed. 

The  further  calculation  of  the  single-phase  induction  motor 
is  identical  with  that  of  the  polyphase  induction  motor,  as  given 
in  the  previous  chapter. 

Often  no  special  motors  are  used  for  single-phase  circuits, 
but  polyphase  motors  adapted  thereto.  An  induction  motor 
with  only  one  primary  winding  could  not  be  started  by  a  phase- 
splitting  device,  and  would  necessarily  be  started  by  external 
means.  A  polyphase  motor,  as  for  instance  a  three-phase  motor 
operating  single-phase,  by  having  two  of  its  terminals  connected 
to  the  single-phase  mains,  is  just  as  satisfactory  a  single-phase 
motor  as  one  built  with  only  one  primary  winding.  The  only 
difference  is  that  in  the  latter  case  a  part  of  the  circumference 
of  the  primary  structure  is  left  without  winding,  while  in  the 
polyphase  motor  this  part  contains  windings  also,  which,  how- 
ever, are  not  used,  or  are  not  effective  when  running  as  single- 
phase  motor,  but  are  necessary  when  starting  by  means  of 
displaced  e.m.fs.  Thus,  in  a  three-phase  motor  operating  from 
single-phase  mains,  in  starting,  the  third  terminal  is  connected 
to  a  phase-displacing  device,  giving  to  the  motor  the  cross  mag- 
netization in  quadrature  to  the  axis  of  the  primary  coil,  which 
at  speed  is  produced  by  the  rotation  of  the  secondary  currents, 
and  which  is  necessary  for  producing  the  torque  by  its  action 
upon  the  secondary  power  currents. 

Thus  the  investigation  of  the  single-phase  induction  motor 
resolves  itself  into  the  investigation  of  the  polyphase  motor 
operating  on  single-phase  circuits. 

2.  LOAD  AND  SPEED  CURVES 

147.  Comparing  thus  a  three-phase  motor  of  exciting  admit- 
tance per  circuit  Y  =  g  —  jb  and  self-inductive  impedances 
ZQ  =  rQ  +  jxQ  and  Zi  =  TI  +  jxi  per  circuit  with  the  same 
motor  operating  as  single-phase  motor  from  one  pair  of  termi- 
nals, the  single-phase  exciting  admittance  is  Y'  =  3  Y  (so  as 
to  give,  the  same  volt-amperes  excitation  3  eF),  the  primary 


330     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

self-inductive  impedance  is  the  same,  ZQ  =  r0  +  jxo',  the  sec- 
ondary self-inductive  impedance  single-phase,  however,  is  only 

y 

Z'i  =  -5-,  since  all  three  secondary  circuits  correspond  to  the 
same  primary  circuit,  and  thus  the  total  impedance  single-phase 

17 

is  Z'  =  ZQ  +  -TT,    while    that    of    the    three-phase    motor    is 

£J     =    ZlQ    "T"    Zl\. 

Assuming  approximately  Z0  =  Zi,  we  have 


Thus,  in  absolute  value, 


s'  =  %  2,  and 
T'  =  2T; 


that  is,  the  characteristic  constant  of  a  motor  running  single- 
phase  is  twice  what  it  is  running  three-phase,  or  polyphase  in 


1000     2000    3000     4000     5000 


7000    3000     9000 


FIG.  181. — Three-phase  induction  motor  on  single-phase  circuit,  load  curves. 

general;  hence,  the  ratio  of  exciting  current  to  current  at  stand- 
still, or  of  waste  flux  to  useful  flux,  is  doubled  by  changing  from 
polyphase  to  single-phase. 

This  explains  the  inferiority  of  the  single-phase  motor  com- 
pared with  the  polyphase  motor. 

As  a  rule,  an  average  polyphase  motor  makes  a  poor  single- 
phase  motor,  and  a  good  single-phase  motor  must  be  an  excellent 
polyphase  motor. 


INDUCTION  MACHINES 


331 


As  instances  are  shown  in  Figs.  181  and  182  the  load  curves 
and  speed  curves  of  the  three-phase  motor  of  which  the  curves  of 
one  circuit  are  given  in  Figs.  176  and  177,  having  the  following 
constants : 

eo  =  110 


Three-phase 

Y    =  0.01  -O.lj, 
ZQ  =  0.1  +  0.3  j, 
Zi  =  0.1  +  0.3,7, 
Thus,  7  =  6.36. 


Single-phase 

Y    =  0.03  -  0.3  j, 

ZQ    =    0.1    +   0.3  J, 

Zl  =  0.033 +. 0.1  j, 
Thus,  7  =  12.72. 


It  is  of  interest  to  compare  Fig.  181  with  Fig.  176  and  to  note 
the  lesser  drop  of  speed  (due  to  the  relatively  lower  secondary 


SLIP,   S"= 


FIG. 


182. — Three-phase   induction   motor   on   single-phase   circuit,    s 

curves. 


resistance)  and  lower  power-factor  and  efficiencies,  especially  at 
light  load.  The  maximum  output  is  reduced  from  3  X  7000  = 
21,000  in  the  three-phase  motor  to  9100  watts  in  the  single-phase 
motor. 

Since,  however,  the  internal  losses  are  less  in  the  single-phase 
motor,  it  can  be  operated  at  from  25  to  30  per  cent,  higher  mag- 
netic density  than  the  same  motor  polyphase,  and  in  this  case  its 
output  is  from  two-thirds  to  three-quarters  that  of  the  poly- 
phase motor. 

148.  The  preceding  discussion  of  the  single-phase  induction 
motor  is  approximate,  and  correct  only  at  or  near  synchronism, 


332     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

where  the  magnetic  field  is  practically  a  uniformly  rotating  field 
of  constant  intensity,  that  is,  the  quadrature  flux  produced  by 
the  armature  magnetization  equal  to  the  main  magnetic  flux 
produced  by  the  impressed  e.m.f. 

If  an  accurate  calculation  of  the  motor  at  intermediate  speed 
and  at  standstill  is  required,  the  changes  of  effective  exciting 
admittance  and  of  secondary  impedance,  due  to  the  decrease 
of  the  quadrature  flux,  have  to  be  considered. 

At  synchronism  the  total  exciting  admittance  gives  the  m.m.f. 
of  main  flux  and  auxiliary  flux,  while  at  standstill  the  quad- 
rature flux  has  disappeared  or  decreased  to  that  given  by  the 
starting  device,  and  thus  the  total  exciting  admittance  has  de- 


1.0  0.9  0.8  0.7  0.6  0.5  0.4  0.3  0.2  0.1  0 

FIG.  183. — Three-phase  induction  motor  on  single-phase  circuit,  torque 

curves. 


creased  to  one-half  of  its  synchronous  value,  or  one-half  plus  the 
exciting  admittance  of  the  starting  flux. 

The  effective  secondary  impedance  at  synchronism  is  the  joint 
impedance  of  all  secondary  circuits;  at  standstill,  however,  only 
the  joint  impedance  of  the  projections  of  the  secondary  coils  on 
the  direction  of  the  main  flux,  that  is,  twice  as  large  as  at  syn- 
chronism. In  other  words,  from  standstill  to  synchronism  the 
effective  secondary  impedance  gradually  decreases  to  one-half 
its  standstill  value  at  synchronism. 

For  fuller  discussion  hereof  the  reader  must  be  referred  to  my 
second  paper  on  the  Single-phase  Induction  Motor,  Transactions 
A.  I.  E.  E.,  1900,  page  37. 

The  torque  in  Fig.  182  obviously  slopes  toward  zero  at  stand- 


INDUCTION  MACHINES  333 

still.  The  effect  of  resistance  inserted  in  the  secondary  of  the 
single-phase  motor  is  similar  to  that  in  the  polyphase  motor  in 
so  far  as  an  increase  of  resistance  lowers  the  speed  at  which  the 
maximum  torque  takes  place.  While,  however,  in  the  poly- 
phase motor  the  maximum  torque  remains  the  same,  and  merely 
shifts  toward  lower  speed  with  the  increase  of  resistance,  in  the 
single-phase  motor  the  maximum  torque  decreases  proportionally 
to  the  speed  at  which  the  maximum  torque  point  occurs,  due  to 
the  factor  (1  —  s)  entering  the  equation  of  the  torque, 

D  =  e2^  (1  -  s). 

Thus,  in  Fig.  183  are  given  the  values  of  torque  of  the  single- 
phase  motor  for  the  same  conditions  and  the  same  motor  of 
which  the  speed  curves  polyphase  are  given  in  Fig.  179. 

The  maximum  value  of  torque  which  can  be  reached  at  any 
speed  lies  on  the  tangent  drawn  from  the  origin  onto  the  torque 
curve  for  7*1  =  0.1  or  short-circuited  secondary.  At  low  speeds 
the  torque  of  the  single-phase  motor  is  greatly  increased  by  the 
insertion  of  secondary  resistance,  just  as  in  the  polyphase  motor. 

3.  STARTING  DEVICES  OF  SINGLE-PHASE  MOTORS 

149.  At  standstill,  the  single-phase  induction  motor  has  no 
starting  torque,  since  the  line  of  polarization  due  to  the  second- 
ary currents  coincides  with  the  axis  of  magnetic  flux  impressed 
by  the  primary  circuit.  Only  when  revolving  is  torque  pro- 
duced, due  to  the  axis  of  secondary  polarization  being  shifted 
by  the  rotation,  against  the  axis  of  magnetism,  until  at  or  near 
synchronism  it  is  in  quadrature  therewith,  and  the  magnetic 
disposition  thus  the  same  as  that  of  the  polyphase  induction 
motor. 

Leaving  out  of  consideration  starting  by  mechanical  means 
and  starting  by  converting  the  motor  into  a  series  or  shunt 
motor,  that  is,  by  passing  the  alternating  current  by  means  of 
commutator  and  brushes  through  both  elements  of  the  motor, 
the  following  methods  of  starting  single-phase  motors  are  left: 

1st.  Shifting  of  the  axis  of  armature  or  secondary  polarization 
against  the  axis  of  generating  magnetism. 

2d.  Shifting  the  axis  of  magnetism,  that  is,  producing  a  mag- 
netic flux  displaced  in  position  from  the  flux  producing  the  arma- 
ture currents. 


334     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

The  first  method  requires  a  secondary  system  which  is  unsym- 
metrical  in  regard  to  the  primary,  and  thus,  since  the  secondary 
is  movable,  requires  means  of  changing  the  secondary  circuit, 
such  as  commutator  brushes  short-circuiting  secondary  coils  in 
the  position  of  effective  torque,  and  open-circuiting  them  in  the 
position  of  opposing  torque. 

Thus  this  method  leads  to  the  repulsion  motor,  which  is  a 
commutator  motor  also. 

With  the  commutatorless  induction  motor,  or  motor  with 
permanently  closed  armature  circuits,  all  starting  devices  con- 
sist in  establishing  an  auxiliary  magnetic  flux  in  phase  with  the 
secondary  currents  in  time,  and  in  quadrature  with  the  line  of 
secondary  polarization  in  space.  They  consist  in  producing  a 
component  of  magnetic  flux  in  quadrature  in  space  with  the 
primary  magnetic  flux  producing  the  secondary  currents,  and 
in  phase  with  the  latter,  that  is,  in  time  quadrature  with  the 
primary  magnetic  flux. 

Thus,  if 

Fp  =  polarization  due  to  the  secondary  currents, 
<i>a  =  auxiliary  magnetic  flux, 

6  =  phase  displacement  in  time  between  3>a  and  3>p, 
and 

T  =  phase  displacement  in  space  between  ^a  and  Fp, 
the  torque  is 

D  =  Fp$a  sin  T  cos  6. 

In  general  the  starting  torque,  apparent  torque  efficiency, 
etc.,  of  the  single-phase  induction  motor  with  any  of  these  de- 
vices are  given  in  per  cent,  of  the  corresponding  values  of  the 
same  motor  with  polyphase  magnetic  flux,  that  is,  with  a  mag- 
netic system  consisting  of  two  equal  magnetic  fluxes  in  quad- 
rature in  time  and  space. 

150.  The  infinite  variety  of  arrangements  proposed  for  start- 
ing single-phase  induction  motors  can  be  grouped  into  three 
classes. 

1.  Phase-splitting  Devices.  The  primary  system  is  composed 
of  two  or  more  circuits  displaced  from  each  other  in  position,  and 
combined  with  impedances  of  different  inductance  factors  so  as 
to  produce  a  phase  displacement  between  them. 

When  using  two  motor  circuits,  they  can  either  be  connected 
in  series  between  the  single-phase*  mains,  and  shunted  with 
impedances  of  different  inductance  factors,  as,  for  instance,  a 


INDUCTION  MACHINES  335 

condensance  and  an  inductance,  or  they  can  be  connected  in 
shunt  between  the  single-phase  mains  but  in  series  with  impe- 
dances of  different  inductance  factors.  Obviously  the  impe- 
dances used  for  displacing  the  phase  of  the  exciting  coils  can 
either  be  external  or  internal,  as  represented  by  high-resistance 
winding  in  one  coil  of  the  motor,  etc. 

In  this  class  belongs  the  use  of  the  transformer  as  a  phase- 
splitting  device  by  inserting  a  transformer  primary  in  series 
with  one  motor  circuit  in  the  main  line  and  connecting  the  other 
motor  circuit  to  the  secondary  of  the  transformer,  or  by  feeding 
one  of  the  motor  circuits  directly  from  the  mains  and  the  other 
from  the  secondary  of  a  transformer  connected  across  the  mains 
with  its  primary.  In  either  case  it  is,  respectively,  the  internal 
impedance,  or  internal  admittance,  of  the  transformer  which  is 
combined  with  one  of  the  motor  circuits  for  displacing  its  phase, 
and  thus  this  arrangement  becomes  most  effective  by  using 
transformers  of  high  internal  impedance  or  admittance  as  con- 
stant power  transformers  or  open  magnetic  circuit  transformers. 

2.  Inductive  Devices.     The  motor  is  excited  by  the  combina- 
tion of  two  or  more  circuits  which  are  in  inductive  relation  to 
each  other.     This  mutual  induction  between  the  motor  circuits 
can  take  place  either  outside  of  the  motor  in  a  separate  phase- 
splitting  device  or  in  the  motor  proper. 

In  the  first  case  the  simplest  form  is  the  divided  circuit  whose 
branches  are  inductively  related  to  each  other  by  passing  around 
the  same  magnetic  circuit  external  to  the  motor. 

In  the  second  case  the  simplest  form  is  the  combination  of  a 
primary  exciting  coil  and  a  short-circuited  secondary  coil  on 
the  primary  member  of  the  motor,  or  a  secondary  coil  closed  by 
an  impedance. 

In  this  class  belong  the  shading  coil  and  the  accelerating  coil. 

3.  Monocyclic  Starting  Devices.     An  essentially  wattless  e.m.f. 
of  displaced  phase  is  produced  outside  of  the  motor,  and  used 
to  energize  a  cross  magnetic  circuit  of  the  motor,  either  directly 
by  a  special  teaser  coil  on  the  motor,  or  indirectly  by  combining 
this  wattless  e.m.f.  with  the  main  e.m.f.  and  thereby  deriving  a 
system  of  e.m.fs.   of  approximately  three-phase  or  any  other 
relation.     In  this   case   the   primary   system  of  the   motor  is 
supplied  essentially  by  a  polyphase  system  of  e.m.fs.  with  a 
single-phase   flow   of   energy,    a   system   which   I   have   called 
"monocyclic." 


336     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

The  wattless  quadrature  e.m.f.  is  generally  produced  by  con- 
necting two  impedances  of  different  inductance  factors  in  series 
between  the  single-phase  mains,  and  joining  the  connection 
between  the  two  impedances  to  the  third  terminal  of  a  three- 
phase  induction  motor,  which  is  connected  with  its  other  two 
terminals  to  the  single-phase  lines,  as  shown  diagrammatically 
in  Fig.  184,  for  a  conductance  a  and  an  inductive  susceptance  -jo,. 

This  starting  device,  when  using  an  inductance  and  a  conden- 
sance  of  proper  size,  can  be  made  to  give  an  apparent  starting 
torque  efficiency  superior  to  that  of  the  polyphase  induction 
motor.  Usually  a  resistance  and  an  inductance  are  used,  which, 
though  not  giving  the  same  starting  torque  efficiency  as  available 
by  the  use  of  a  condensance,  have  the  advantage  of  greater 
simplicity  and  cheapness.  After  starting,  the  impedances  are 
disconnected. 

For  a  complete  discussion  and  theoretical  investigation  of  the 


FIG.  184. — Connections  for  starting  single-phase  motor. 

different  starting  devices,  the  reader  must  be  referred  to  the 
paper  on  the  single-phase  induction  motor,  American  Institute 
of  Electrical  Engineers'  Transactions,  February,  1898." 

151.  The  use  of  the  resistance-inductance,  or  monocyclic, 
starting  device  with  three-phase  wound  induction  motor  will 
be  discussed  somewhat  more  explicitly  as  the  only  method  not  us- 
ing condensers  which  has  found  extensive  commercial  application. 
It  gives  relatively  the  best  starting  torque  and  torque  efficiencies. 

In  Fig.  184,  M  represents  a  three-phase  induction  motor  of 
which  two  terminals,  1  and  2,  are  connected  to  single-phase  mains 
and  the  terminal  3  to  the  common  connection  of  a  conduct- 
ance a  (that  is,  a  resistance  - j  and  an  equal  susceptance  —  ja 

(thus  a  reactance  H — )  connected  in  series  across  the  mains. 
Let  Y  =  g  —  jb  =  total  admittance  of  motor  between  termi- 


INDUCTION  MACHINES  337 

nals  1  and  2  while  at  rest.  We  then  have  HY  =  total  admit- 
tance from  terminal  3  to  terminals  1  and  2,  regardless  of  whether 
the  motor  is  delta-  or  F-wound. 

If  e  =  e.m.f.  in  the  single-phase  mains  and  E  =  difference  of 
potential  across  conductance  a  of  the  starting  device,  then  we 
have  the  current  in  a  as  /i  =  Ea, 

and  the  e.m.f.  across  —  ja  as    e  —  E' 
thus,  the  current  in  —  ja  is 

li  =  -  ja  (e  -  E), 

and  the  current  in  the  cross  magnetizing  motor  circuit  from  3  to 
1,  2  is 

/o  =  /i  -  /2  =  Ea  +  ja  (e  -  E). 

The  e.m.f.  ^0  of  the  cross  magnetizing  circuit  is,  as  may  be  seen 
from  the  diagram  of  e.m.fs.,  which  form  a  triangle  with  e,  E  and 
e  —  E  as  sides, 

Eo  =  E  -  (e  -  E)  =  2  E  -  e, 
and  since 

!»  =  H  YEQ, 
we  have 

Ea  +  ja  (e  -  E)  =  %  Y  (2E  -  e). 

This  expression  solved  for  E  becomes 

€ 


which  from  the  foregoing  value  of  EQ  gives 

3eaQ'+l)     . 
~3a-3ja-8F' 
or,  substituting 

Y-g-  jb, 

expanding,  and  multiplying  both  numerator  and  denominator  by 
(3  a -80)  +j(3a  -  86), 


gives  EQ  =  ea 


and  the  imaginary  component  thereof,  or  e.m.f.  in  quadrature 
to  e  in  time  and  in  space,  is 


338     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

In  the  same  motor  on  a  three-phase  circuit  this  quadrature 
e.m.f.  is  the  altitude  of  the  equilateral  triangle  with  e  as  sides, 

thus  =  —  je  — 7r-,  and  since  the  starting  torque  of  the  motor  is  pro- 
portional to  this  quadrature  e.m.f.,  the  relative  starting  torque 
of  the  monocyclic  starting  device,  or  the  ratio  of  starting  torque 
of  the  motor  with  monocyclic  starting  device  to  that  of  the 
same  motor  on  three-phase  circuit,  is 

/)/  =  •  EG!_  :     2a  2a- %(</-&) 


A  starting  device  which  has  been  extensively  used  is  the 
condenser  in  the  tertiary  circuit.  In  its  usual  form  it  can  be 
considered  as  a  modification  of  the  monocyclic  starting  device, 
by  using  a  condensance  as  the  one  impedance  and  making  the 
other  impedance  infinite,  that  is,  omitting  it.  It  thus  comprises 
a  three-phase  induction  motor,  in  which  two  terminals  are  con- 
nected to  the  single-phase  supply  and  the  third  terminal  and 
one  of  the  main  terminals  to  a  condenser.  Usually  the  .con- 
denser is  left  in  circuit  after  starting,  and  made  of  such  size 
that  its  leading  current  compensates  for  the  lagging  magnetizing 
current  of  the  motor,  and  the  motor  thus  gives  approximately 
unity  power-factor. 

For  further  discussion  of  this  subject  the  reader  is  referred  to 
the  paper  on  "  Single-phase  Induction  Motors,"  mentioned  above, 
and  to  the  "  Theory  and  Calculation  of  Alternating-current  Phe- 
nomena" and  "Theory  and  Calculation  of  Electrical  Apparatus." 

4.  ACCELERATION  WITH  STARTING  DEVICE 

152.  The  torque  of  the  single-phase  induction  motor  (without 
a  starting  device)  is  proportional  to  the  product  of  main  flux,  or 
magnetic  flux  produced  by  the  primary  impressed  e.m.f.,  and 
the  speed.  Thus  it  is  the  same  as  in  the  polyphase  motor  at  or 
very  near  synchronism,  but  falls  off  with  decreasing  speed  and 
becomes  zero  at  standstill. 

To  produce  a  starting  torque,  a  device  has  to  be  used  to  impress 
an  auxiliary  magnetic  flux  upon  the  motor,  in  quadrature  with 
the  main  flux  in  time  and  in  space,  and  the  starting  torque 
is  proportional  to  this  auxiliary  or  quadrature  flux.  During 
acceleration  or  at  intermediate  speed  the  torque  of  the  motor  is 


INDUCTION  MACHINES  339 

the  resultant  of  the  main  torque,  or  torque  produced  by  the  pri- 
mary main  flux,  and  the  auxiliary  torque  produced  by  the  auxil- 
iary quadrature  or  starting  flux.  In  general,  this  resultant  torque 
is  not  the  sum  of  main  and  auxiliary  torque,  but  often  less,  due 
to  the  interaction  between  the  motor  and  the  starting  device. 

Most  starting  devices  depend  more  or  less  upon  the  total 
admittance  of  the  motor  and  its  power-factor.  With  increasing 
speed,  however,  the  total  admittance  of  the  motor  decreases 
and  its  power-factor  increases,  and  an  auxiliary  torque  device 
suited  for  the  admittance  of  the  motor  at  standstill  will  not  be 
suited  for  the  changed  admittance  at  speed. 

The  currents  produced  in  the  secondary  by  the  main  or  pri- 
mary magnetic  flux  are  carried  by  the  rotation  of  the  motor  more 
or  less  into  quadrature  position,  and  thus  produce  the  quad- 
rature flux  giving  the  main  torque  as  discussed  before. 

This  quadrature  component  of  the  main  flux  generates  an  e.m.f . 
in  the  auxiliary  circuit  of  the  starting  device,  and  thus  changes 
the  distribution  of  currents  and  e.m.fs.  in  the  starting  device. 
The  Circuits  of  the  starting  device  then  contain,  besides  the  motor 
admittance  and  external  admittance,  an  active  counter  e.m.f., 
changing  with  the  speed.  Inversely,  the  currents  produced  by 
the  counter  e.m.f.  of  the  motor  in  the  auxiliary  circuit  react  upon 
the  counter  e.m.f.,  that  is,  upon  the  quadrature  component  or 
main  flux,  and  change  it. 

Thus  during  acceleration  we  have  to  consider — 

1.  The  effect  of  the  change  of  total  motor  admittance  and  its 
power-factor  upon  the  starting  device. 

2.  The  effect  of  the  counter  e.m.f.  of  the  motor  upon  the  start- 
ing device  and  the  effect  of  the  starting  device  upon  the  counter 
e.m.f.  of  the  motor. 

1.  The  total  motor  admittance  and  its  power-factor  change 
very  much  during  acceleration  in  motors  with  short-circuited 
low-resistance  secondary.  In  such  motors  the  admittance  at 
rest  is  very  large  and  its  power-factor  low,  and  with  increasing 
speed  the  admittance  decreases  and  its  power-factor  increases 
greatly.  In  motors  with  short-circuited  high-resistance  second- 
ary the  admittance  also  decreases  greatly  during  acceleration, 
but  its  power-factor  changes  less,  being  already  high  at  stand- 
still. Thus  the  starting  device  will  be  affected  less,  Such 
motors,  however,  are  inefficient  at  speed.  In  motors  with  varia- 
ble secondary  resistance  the  admittance  and  its  power-factor 


340     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

can  be  maintained  constant  during  acceleration  by  decreasing 
the  resistance  of  the  secondary  circuit  in  correspondence  with  the 
increasing  counter  e.m.f.  Hence,  in  such  motors  the  starting 
device  is  not  thrown  out  of  adjustment  by  the  changing  admit- 
tance during  acceleration. 

In  the  phase-splitting  devices,  and  still  more  in  the  inductive 
devices,  the  starting  torque  depends  upon  the  internal  or  motor 
admittance,  and  is  thus  essentially  affected  by  the  change  of 
admittance  during  acceleration,  and  by  the  appearance  of  a 
counter  e.m.f.  during  acceleration,  which  throws  the  starting 
device  out  of  its  proper  adjustment,  so  that  in  some  cases  while  a 
considerable  torque  exists  at  standstill,  this  torque  becomes  zero 
and  then  reverses  at  some  intermediate  speed,  and  the  motor, 
while  starting  with  fair  torque,  is  not  able  to  run  up  to  speed 
with  the  starting  device  in  circuit.  Especially  is  this  the  case 
where  capacity  is  used  in  the  starting  device.  With  the  mono- 
cyclic  starting  this  effect  is  small  in  any  case  and  absent  when  a 
condenser  is  used  in  the  tertiary  circuit,  and  therefore  the 
latter  may  advantageously  be  left  in  the  circuit  at  speed. 

IV.  Induction  Generator 

1.  INTRODUCTION 

163.  In  the  range  of  slip  from  s  =  0  to  s  =  1,  that  is,  from 
synchronism  to  standstill,  torque,  power  output,  and  power 
input  of  the  induction  machine  are  positive,  and  the  machine 
thus  acts  as  a  motor,  as  discussed  before. 

Substituting,  however,  in  the  equations  in  paragraph  1  for 
s  values  >  1,  corresponding  to  backward  rotation  of  the  ma- 
chine, the  power  input  remains  positive,  the  torque  also  remains 
positive,  that  is,  in  the  same  direction  as  for  s  <  1 ;  but  since  the 
speed  (I  —  s)  becomes  negative  or  in  opposite  direction,  the 
power  output  is  negative,  that  is,  the  torque  in  opposite  direc- 
tion to  the  speed.  In  this  case  the  machine  consumes  electrical 
energy  in  its  primary  and  mechanical  energy  by  a  torque  oppos- 
ing the  rotation,  thus  acting  as  brake. 

The  total  power,  electrical  as  well  as  mechanical,  is  con- 
sumed by  internal  losses  of  the  motor.  Since,  however,  with 
large  slip  in  a  low-resistance  motor  the  torque  and  power  are 
small,  the  braking  power  of  the  induction  machine  at  backward 


INDUCTION  MACHINES 


341 


rotation  is,  as  a  rule,  not  considerable,  excepting  when  using 
high  resistance  in  the  armature  circuit. 


Z0«  Zj-  0.1+  0.3  j     Y  -  0.01  -  0.1  J 
110  VOLTS  CONSTANT  FREQUENCY 


-1000         -2000        -3000        -4000        -5000        -6000         -7000        -8000        -9000 

FIG.  185. — Induction  generator  load  curves. 


TORQUE  POWER 
8000 


_0 

loo; 

-4000 


-6000 


113    1.2    1U    1009    0,8    0:7    o!e    0  5    0!4    Oi3    ol2    0. 


ACTI 

A    ^ 


SLIP  FACTION  OF  SYNCHROS  SM 


CONSTANT  FREQUENCY 

CONSTANT  TERMINAL 

VOLTAGE  OF  110 


Z0- 

Y  -  0.01  - 


0.4    05    060 


160 


140 


100. 


FIG.  186. — Induction  machine  speed  curves. 

Substituting  for  s  negative  values,  corresponding  to  a  speed 
above  synchronism,  torque  and  power  output  and  power  input 


342     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

become  negative,  and  a  load  curve  can  be  plotted  for  the  induc- 
tion generator  which  is  very  similar,  but  the  negative  counter- 
part of  the  induction  motor  load  curve.  It  is  for  the  machine 
shown  as  motor  in  Fig.  176  given  as  Fig.  185,  while  Fig.  186 
gives  the  complete  speed  curve  of  this  machine  from  s  =  1.5  to 
*  =  -1. 

The  generator  part  of  the  curve,  for  s  <  0,  is  of  the  same  char- 
acter as  the  motor  part,  s  >  0,  but  the  maximum  torque  and 
maximum  output  of  the  machine  as  generator  are  greater  than 
as  motor. 

Thus  an  induction  motor  when  speeded  up  above  synchronism 
acts  as  a  powerful  brake  by  returning  energy  into  the  lines,  and 
the  maximum  braking  effort  and  also  the  maximum  electric 
power  returned  by  the  machine  will  be  greater  than  the  maxi- 
mum motor  torque  or  output. 

2.  CONSTANT-SPEED  INDUCTION  OR  ASYNCHRONOUS 
GENERATOR 

154.  The  curves  in  Fig.  185  are  calculated  at  constant  fre- 
quency /,  and  thus  to  vary  the  output  of  the  machine  as  gen- 
erator the  speed  has  to  be  increased.  This  condition  may  be 
realized  in  case  of  induction  generators  running  in  parallel  with 
synchronous  generators  under  conditions  where  it  is  desirable 
that  the  former  should  take  as  much  load  as  its  driving  power 
permits;  as,  for  instance,  if  the  induction  generator  is  driven  by 
a  water  power  while  the  synchronous  generator  is  driven  by 
a  steam  engine.  In  this  case  the  control  of  speed  would  be 
effected  on  the  synchronous  generator,  and  the  induction  gen- 
erator be  without  speed-controlling  devices,  running  up  beyond 
synchronous  speed  as  much  as  required  to  consume  the  power 
supplied  to  it. 

Conversely,  however,  if  an  induction  machine  is  driven  at 
constant  speed  and  connected  to  a  suitable  circuit  as  load,  the 
frequency  given  by  the  machine  will  not  be  synchronous  with 
the  speed,  or  constant  at  all  loads,  but  decreases  with  increasing 
load  from  practically  synchronism  at  no  load,  and  thus  for  the 
induction  generator  at  constant  speed  a  load  curve  can  be  con- 
structed as  shown  in  Fig.  187,  giving  the  decrease  of  frequency 
with  increasing  load  in  the  same  manner  as  the  speed  of  the 
induction  motor  at  constant  frequency  decreases  with  the  load. 
In  the  calculation  of  these  induction  generator  curves  for  con- 


INDUCTION  MACHINES 


343 


slant  speed  the  change  of  frequency  with  the  load  has  obviously 
to  be  considered,  that  is,  in  the  equations  the  reactance  x0  has 
to  be  replaced  by  the  reactance  XQ  (1  —  s),  otherwise  the  equa- 
tions remain  the  same. 


FIG.  187. — Induction  generator  load  curves. 


3.  POWER-FACTOR  OF  INDUCTION  GENERATOR 

155.  The  induction  generator  differs  'essentially  from  a  syn- 
chronous alternator  (that  is,  a  machine  in  which  an  armature 
revolves  relatively  through  a  constant  or  continuous  magnetic 
field)  by  having  a  power-factor  requiring  leading  current ;  that 
is,  in  the  synchronous  alternator  the  phase  relation  between 
current  and  terminal  voltage  depends  entirely  upon  the  external 
circuit,  and  according  to  the  nature  of  the  circuit  connected  to 
the  synchronous  alternator  the  current  can  lag  or  lead  the  ter- 
minal voltage  or  be  in  phase  therewith.  In  the  induction  or 
asynchronous  generator,  however,  the  current  must  lead  the  ter- 
minal voltage  by  the  angle  corresponding  to  the  load  and  voltage 
of  the  machine,  or,  in  other  words,  the  phase  relation  between 
current  and  voltage  in  the  external  circuit  must  be  such  as 
required  by  the  induction  generator  at  that  particular  load. 

Induction  generators  can  operate  only  on  circuits  with  lead- 
ing current  or  circuits  of  negative  effective  reactance. 


344     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

In  Fig.  188  are  given  for  the  constant-speed  induction  gen- 
erator in  Fig.  230  as  function  of  the  impedance  of  the  external 

circuit  z  =  -?•   as  abscissas  (where  eQ  =  terminal  voltage,  iQ  = 

2o 

current  in  external  circuit),  the  leading  power-factor  p  =  cos  6 
required  in  the  load,  the  inductance  factor  q  =  sin  6,  and  the 
frequency. 

Hence,  when  connected  to  a  circuit  of  impedance  z  this  induc- 
tion generator  can  operate  only  if  the  power-factor  of  its  circuit 
is  p',  and  if  this  is  the  case  the  voltage  is  indefinite,  that  is,  the 
circuit  unstable,  even  neglecting  the  impossibility  of  securing 
exact  equality  of  the  power-factor  of  the  external  circuit  with 
that  of  the  induction  generator. 


FIG.  188. — Three-phase  induction  generator  power  factor  and  inductance 
factor  of  external  circuit. 

Two  possibilities  thus  exist  with  such  an  induction  generator 
circuit. 

1st.  The  power-factor  of  the  external  circuit  is  constant  and 
independent  of  the  voltage,  as  when  the  external  circuit  consists 
of  resistances,  inductances,  and  capacities. 

In  this  case  if  the  power-factor  of  the  external  circuit  is  higher 
than  that  of  the  induction  generator,  that  is,  the  leading  current 
less,  the  induction  generator  fails  to  excite  and  generate.  If  the 
power-factor  of  the  external  circuit  is  lower  than  that  of  the 
induction  generator,  the  latter  excites  and  its  voltage  rises  until 
by  saturation  of  its  magnetic  circuit  and  the  consequent  increase 
of  exciting  admittance,  that  is,  decrease  of  internal  power-factor, 
its  power-factor  has  fallen  to  equality  with  that  of  the  external 
circuit. 


INDUCTION  MACHINES 


345 


In  this  respect  the  induction  generator  acts  like  the  direct- 
current  shunt  generator,  and  gives  load  characteristics  very 
similar  to  those  of  the  direct-current  shunt  generator  as  dis- 
cussed in  B;  that  is,  it  becomes  stable  only  at  saturation,  but 


.Z^O.I  +  O.Sjl     ATFULL 

V«0.01—  O.lj    I  FREQUENCY 


FIG.  189. — Induction  generator  and  synchronous  motor  load  curves. 

loses  its  excitation  and  thus  drops  its  load  as  soon  as  the  voltage 
falls  below  saturation. 

Since,  however,  the  field  of  the  induction  generator  is  alter- 
nating, it  is  usually  not  feasible  to  run  at  saturation,  due  to  ex- 
cessive hysteresis  losses,  except  for  very  low  frequencies. 


346     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

2d.  The  power-factor  of  the  external  circuit  depends  upon  the 
voltage  impressed  upon  it. 

This,  for  instance,  is  the  case  if  the  circuit  consists  of  a  syn- 
chronous motor  or  contains  synchronous  motors  or  synchronous 
converters. 

In  the  synchronous  motor  the  current  is  in  phase  with  the 
impressed  e.m.f.  if  the  impressed  e.m.f.  equals  the  counter  e.m.f. 
of  the  motor  plus  the  internal  loss  of  voltage.  It  is  leading  if  the 
impressed  e.m.f.  is  less,  and  lagging  if  the  impressed  e.m.f.  is 
more.  Thus  when  connecting  an  induction  generator  with  a 
synchronous  motor,  at  constant  field  excitation  of  the  latter  the 


01        02        0,3        04        05        06        017        0.8       019        IjO         11.        12       13        14        lf.5 


FIG.  190. — Induction  generator  and  synchronous  converter,  phase  control, 

no  line  impedance. 

voltage  of  the  induction  generator  rises  until  it  is  as  much  below 
the  counter  e.m.f.  of  the  synchronous  motor  as  required  to  give 
the  leading  current  corresponding  to  the  power-factor  of  the 
generator.  Thus  a  system  consisting  of  a  constant-speed  induc- 
tion generator  and  a  synchronous  motor  at  constant  field  excita- 
tion is  absolutely  stable.  At  constant  field  excitation  of  the 
synchronous  motor",  at  no  load  the  synchronous  motor  runs 
practically  at  synchronism  with  the  induction  generator,  with  a 
terminal  voltage  slightly  below  the  counter  e.m.f.  of  the  syn- 
chronous motor.  With  increase  of  load  the  frequency  and  thus 
the  speed  of  the  synchronous  motor  drops,  due  to  the  slip  of 
frequency  in  the  induction  generator,  and  the  voltage  drops, 


INDUCTION  MACHINES  347 

due  to  the  increase  of  leading  current  required  and  the  decrease 
of  counter  e.m.f.  caused  by  the  decrease  of  frequency. 

By  increasing  the  field  excitation  of  the  synchronous  motor 
with  increase  of  load,  obviously  the  voltage  of  the  generator  can 
be  maintained  constant,  or  even  increased  with  the  load. 

When  running  from  an  induction  generator,  a  synchronous 
motor  gives  a  load  curve  very  similar  to  the  load  curve  of  an 
induction  motor  running  from  a  synchronous  generator;  that  is, 
a  magnetizing  current  at  no  load  and  a  speed  gradually  decreas- 
ing with  the  increase  of  load  up  to  a  maximum  output  point,  at 
which  the  speed  curve  bends  sharply  down,  the  current  curve 
upward,  and  the  motor  drops  out  of  step. 

The  current,  however,  in  the  case  of  the  synchronous  motor 
operated  from  an  induction  generator  is  leading,  while  it  is  lag- 
ging in  an  induction  motor  operated  from  a  synchronous  genera- 
tor. In  either  case  it  demagnetizes  the  synchronous  machine 
and  magnetizes  the  induction  machine,  that  is,  the  synchronous 
machine  supplies  magnetization  to  the  induction  machine. 

In  Fig.  189  is  shown  the  load  curve  of  a  synchronous  motor 
operated  from  the  induction  generator  in  Fig.  187. 

In  Fig.  190  is  shown  the  load  curve  of  an  over-compounded 
synchronous  converter  operated  from  an  induction  generator, 
the  over-compounding  being  such  as  to  give  approximately 
constant  terminal  voltage  e. 

156.  Obviously  when  operating  a  self-exciting  synchronous 
converter  from  an  induction  generator  the  system  is  unstable 
also;  if  both  machines  are  below  magnetic  saturation,  since  in 
this  case  in  both  machines  the  generated  e.m,f.  is  proportional 
to  the  field  excitation  and  the  field  excitation  proportional  to 
the  voltage;  that  is,  with  an  unsaturated  induction  generator  the 
synchronous  converter  operated  therefrom  must  have  its  mag- 
netic field  excited  to  a  density  above  the  bend  of  the  saturation 
curve. 

Since  the  induction  generator  requires  for  its  operation  a  circuit 
with  leading  current  varying  with  the  load  in  the  manner  de- 
termined by  the  internal  constants  of  the  motor,  to  make  an 
induction  or  asynchronous  generator  suitable  for  operation  on  a 
general  alternating-current  circuit,  it  is  necessary  to  have  a  syn- 
chronous machine  as  exciter  in  the  circuit  consuming  leading 
current,  that  is,  supplying  the  required  lagging  or  magnetizing 
current  to  the  induction  generator;  and  in  this  case  the  voltage 


348     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

of  the  system  is  controlled  by  the  field  excitation  of  the  syn- 
chronous machine,  that  is,  its  counter  e.m.f.  Either  a  synchro- 
nous motor  of  suitable  size  running  light  can  be  used  herefor  as 
exciter  of  the  induction  generator,  or  the  exciting  current  of  the 
induction  generator  may  be  derived  from  synchronous  motors  or 
converters  in  the  same  system,  or  from  synchronous  alternating- 
current  generators  operated  in  parallel  with  the  induction  gen- 
erator, in  which  latter  case,  however,  these  currents  can  be  said 
to  come  from  the  synchronous  alternator  as  lagging  currents. 
Electrostatic  condensers,  as  an  underground  cable  system,  may 
also  be  used  for  excitation,  but  in  this  case  besides  the  condensers  a 
synchronous  machine  or  other  means  is  required  to  secure  stability. 

The  induction  machine  may  thus  be  considered  as  consuming 
a  lagging  reactive  magnetizing  current  at  all  speeds,  and  con- 
suming a  power  current  below  synchronism,  as  motor,  supplying 
a  power  current  (that  is,  consuming  a  negative  power  current) 
above  synchronism,  as  generator. 

Therefore,  induction  generators  are  best  suited  for  circuits 
which  normally  carry  leading  currents,  as  synchronous  motor 
and  synchronous  converter  circuits,  but  less  suitable  for  circuits 
with  lagging  currents,  since  in  the  latter  case  an  additional  syn- 
chronous machine  is  required,  giving  all  the  lagging  currents 
of  the  system  plus  the  induction  generator  exciting  current. 

Obviously,  when  running  induction  generators  in  parallel  with 
a  synchronous  alternator  no  synchronizing  is  required,  but  the 
induction  generator  takes  a  load  corresponding  to  the  excess  of 
its  speed  over  synchronism,  or  conversely,  if  the  driving  power 
behind  the  induction  generator  is  limited,  no  speed  regulation 
is  required,  but  the  induction  generator  runs  at  a  speed  exceeding 
synchronism  by  the  amount  required  to  consume  the  driving 
power. 

The  foregoing  consideration  obviously  applies  to  the  polyphase 
induction  generator  as  well  as  to  the  single-phase  induction 
generator,  the  latter,  however,  requiring  a  larger  exciter  in  con- 
sequence of  its  lower  power-factor.  Therefore,  even  in  a  single- 
phase  induction  generator,  preferably  polyphase  excitation  is 
used,  that  is,  the  induction  machine  and  its  synchronous  exciter 
wound  as  polyphase  machines,  but  the  load  connected  to  one 
phase  only  of  the  induction  machine.  The  curves  shown  in  the 
preceding  apply  to  the  machine  as  polyphase  generator. 

The  effect  of  resistance  in  the  secondary  is  essentially  the 


INDUCTION  MACHINES  349 

same  in  the  induction  generator  as  in  the  induction  motor.  An 
increase  of  resistance  increases  the  slip,  that  is,  requires  an  in- 
crease of  speed  at  the  same  torque,  current,  and  output,  and  thus 
correspondingly  lowers  the  efficiency. 

Induction  generators  have  been  proposed  and  used  to  some 
extent  for  high-speed  prime  movers,  as  steam  turbines,  since 
their  squirrel-cage  rotor  appears  mechanically  better  suited  for 
very  high  speeds  than  the  revolving  field  of  the  synchronous 
generator. 

The  foremost  use  of  induction  generators  will  probably  be  for 
collecting  small  water  powers  in  one  large  system,  due  to  the  far 
greater  simplicity,  reliability,  and  cheapness  of  a  small  induction 
generator  station  feeding  into  a  big  system  compared  with  a 
small  synchronous  generator  station.  The  induction  generator 
station  requires  only  the  hydraulic  turbine,  the  induction  ma- 
chine, and  the  step-up  transformer,  but  does  not  even  require  a 
turbine  governor,  and  so  needs  practically  no  attention,  as  the 
control  of  voltage,  speed,  and  frequency  takes  place  by  a  syn- 
chronous generator  or  motor  main  station,  which  collects  the 
power  while  the  individual  induction  generator  stations  feed  into 
the  system  as  much  power  as  the  available  water  happens  to 
supply. 

The  synchronous  induction  motor,  comprising  a  single-phase 
or  polyphase  primary  and  a  single-phase  secondary,  tends  to 
drop  into  synchronism  and  then  operates  essentially  as  reaction 
machine.  A  number  of  types  of  synchronous  induction  genera- 
tors have  been  devised,  either  with  commutator  for  excitation 
or  without  commutator  and  with  excitation  by  low-frequency 
synchronous  or  commutating  machine,  in  the  armature,  or  by 
high-frequency  excitation.  For  particulars  regarding  these  very 
interesting  machines,  see  "  Theory  and  Calculation  of  Alternat- 
ing-current Phenomena." 

V.  Induction  Booster 

157.  In  the  induction  machine,  at  a  given  slip  s,  current  and 
terminal  voltage  are  proportional  to  each  other  and  of  constant 
phase  relation,  and  their  ratio  is  a  constant.  Thus  when  con- 
nected in  an  alternating-current  circuit,  whether  in  shunt  or  in 
series,  and  held  at  a  speed  giving  a  constant  and  definite  slip  s, 
either  positive  or  negative,  the  induction  machine  acts  like  a 
constant  impedance. 


350     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

The  apparent  impedance  and  its  components,  the  apparent 
resistance  and  apparent  reactance  represented  by  the  induction 
machine,  vary  with  the  slip.  At  synchronism  apparent  impe- 
dance, resistance,  and  reactance  are  a  maximum.  They  decrease 
with  increasing  positive  slip.  With  increasing  negative  slip  the 
apparent  impedance  and  reactance  decrease  also,  the  apparent 


FIG.  191. — Effective  impedance  of  three-phase  induction  machine. 

resistance  decreases  to  zero  and  then  increases  again  in  negative 
direction  as  shown  in  Fig.  191,  which  gives  the  apparent  impe- 
dance, resistance,  and  reactance  of  the  machine  shown  in  Figs, 
176  and  177,  etc.,  with  the  speed  as  abscissas. 

The  cause  is  that  the  power  current  is  in  opposition  to  the  ter- 
minal voltage  above  synchronism,  and  thereby  the  induction 


INDUCTION  MACHINES  351 

machine  behaves  as  an  impedance  of  negative  resistance,  that  is, 
adding  a  power  e.m.f.  into  the  circuit  proportional  to  the  current. 

As  may  be  seen  herefrom,  the  induction  machine  when  inserted 
in  series  in  an  alternating-current  circuit  can  be  used  as  a  booster, 
that  is,  as  an  apparatus  to  generate  and  insert  in  the  circuit  an 
e.m.f.  proportional  to  the  current,  and  the  amount  of  the  boosting 
effect  can  be  varied  by  varying  the  speed,  that  is,  the  slip  at 
which  the  induction  machine  is  revolving.  Above  synchronism 
the  induction  machine  boosts,  that  is,  raises  the  voltage;  below 
synchronism  it  lowers  the  voltage;  in  either  case  also  adding  an 
out-of-phas.e  e.m.f.  due  to  its  reactance.  The  greater  the  slip, 
either  positive  or  negative,  the  less  is  the  apparent  resistance, 
positive  or  negative,  of  the  induction  machine. 

The  effect  of  resistance  inserted  in  the  secondary  of  the  induc- 
tion booster  is  similar  to  that  in  the  other  applications  of  the 
induction  machine;  that  is,  it  increases  the  slip  required  for  a 
certain  value  of  apparent  resistance,  thereby  lowering  the  effi- 
ciency of  the  apparatus,  but  at  the  same  time  making  it  less  de- 
pendent upon  minor  variations  of  speed ;  that  is,  requires  a  lesser 
constancy  of  slip,  and  thus  of  speed  and  frequency,  to  give  a 
steady  boosting  effect. 

VI.  Phase  Converter 

158.  It  may  be  seen  from  the  preceding  that  the  induction 
machine  can  operate  equally  well  as  motor,  below  synchronism, 
and  as  generator,  above  synchronism. 

In  the  single-phase  induction  machine  the  motor  or  generator 
action  occurs  in  one  primary  circuit  only,  but  in  the  direction  in 
quadrature  to  the  primary  circuit  there  is  a  mere  magnetizing 
current  either  in  the  secondary,  in  the  single-phase  motor  proper, 
or  in  an  auxiliary  field-circuit,  in  the  monocyclic  motor. 

The  motor  and  generator  action  can  occur,  however,  simul- 
taneously in  the  same  machine,  some  of  the  primary  circuits 
acting  as  motor,  others  as  generator  circuits.  Thus,  if  one  of 
the  two  circuits  of  a  quarter-phase  induction  machine  is  con- 
nected to  a  single-phase  system,  in  the  second  circuit  an  e.m.f.  is 
generated  in  quadrature  with  and  equal  to  the  generated  e.m.f. 
in  the  first  circuit;  and  this  e.m.f.  can  thus  be  utilized  to  produce 
currents  which,  with  currents  taken  from  the  primary  single- 
phase  mains,  give  a  quarter-phase  system.  Or,  in  a  three-phase 
motor  connected  with  two  of  its  terminals  to  a  single-phase  sys- 


352     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

tern,  from  the  third  terminal  an  e.m.f.  can  be  derived  which, 
with  the  single-phase  system  feeding  the  induction  machine,  com- 
bines to  a  three-phase  system.  The  induction  machine  in  this 
application  represents  a  phase  converter. 

The  phase  converter  obviously  combines  the  features  of  a 
single-phase  induction  motor  with  those  of  a  double  transformer, 
transformation  occurring  from  the  primary  or  motor  circuit  to 
the  secondary  or  armature,  and  from  the  secondary  to  the  ter- 
tiary or  generator  circuit. 

Thus,  in  a  quarter-phase  motor  connected  to  single-phase  mains 
with  one  of  its  circuits,  if 

Y  =  g  —  jb  =  primary  polyphase  exciting  admittance, 
ZQ  =  TQ  -f-  JXQ  =  self  -inductive  impedance  per  primary  or  ter- 

tiary circuit, 
Zi  =  ri  +  jxi  =  resultant  single-phase  self-inductive  impe- 

dance of  secondary  circuits. 
Let 

e  =  e.m.f.  generated  by  the  mutual  flux  and 
Z  =  r  +  jx  =  impedance  of  the  external  circuit  supplied  by 
the  phase  converter  as  generator  of  second  phase. 

We  then  have 

/> 

I  =  71?    —  current  of  second  phase  produced  by  phase 

Zr  T  Z»o 

converter, 
E  —  IZ  =  „  .    „    =  -   —  ;=-  =  terminal  voltage  at  genera- 

&  -\-  LQ         1          L  o 

~~z 

tor  circuit  of  phase  converter. 
The  current  in  the  secondary  of  the  phase  converter  is  then 

/!    =   /  +  /'+  I", 

where 

^ 

I  =  load  current  =  ~      „ 


I'  =  eY  =  exciting  current  of  quadrature  magnetic  flux, 

€S 

I'  =  -    ;  —  :  —  =  current  required  to  revolve  the  machi 
ri+jsxi 

and  the  primary  current  is 

?•'-&>  !', 

where 

/'  =  eY  =  exciting  current  of  main  magnetic  flux. 


INDUCTION  MACHINES  353 

From  these  currents  the  e.m.fs.  are  derived  in  a  similar  manner 
as  in  the  induction  motor  or  generator. 

Due  to  the  internal  losses  in  the  phase  converter,  the  e.m.fs. 
of  the  two  circuits,  the  motor  and  generator  circuits,  are  prac- 
tically in  quadrature  with  each  other  and  equal  only  at  no  load, 
but  shift  out  of  phase  and  become  more  unequal  with  increase  of 
load,  the  unbalancing  depending  upon  the  constants  of  the  phase 
converter. 

An  interesting  application  of  the  phase  converter  is  made 
in  single-phase  induction  motor  railroading.  In  this,  the 
phase  converter  is  connected  in  series  to  the  induction  motor 
which  drives  the  car.  This  avoids  the  increase  of  unbalanc- 
ing of  the  phases  with  increase  of  load,  and  makes  it  possi- 
ble by  properly  connected  series  transformers  to  maintain 
perfect  phase  and  voltage  balance  on  the  driving  motor. 
Usually,  a  quarter-phase  phase  converter  and  quarter-phase 
induction  motor  is  used,  and  the  motor  phase  of  the  phase 
converter  is  connected  in  series  to  one  of  the  phases  of  the 
motor  into  the  single-phase  supply  circuit,  while  the  genera- 
tor phase  of  the  phase  converter  feeds  the  other  phase  of  the 
driving  motor. 

It  is  obvious  that  the  induction  machine  is  used  as  phase  con- 
verter only  to  change  single-phase  to  polyphase,  since  a  change 
from  one  polyphase  system  to  another  polyphase  system  can  be 
effected  by  stationary  transformers.  A  change  from  single- 
phase  to  polyphase,  however,  requires  a  storage  of  energy,  since 
the  power  arrives  as  single-phase  pulsating,  and  leaves  as  steady 
polyphase  flow,  and  the  momentum  of  the  revolving  phase  con- 
verter secondary  stores  and  returns  the  energy. 

With  increasing  load  on  the  generator  circuit  of  the  phase 
converter  its  slip  increases,  but  less  than  with  the  same  load  as 
mechanical  output  from  the  machine  as  induction  motor. 

An  application  of  the  phase  converter  is  made  in  single-phase 
motors  by  closing  the  tertiary  or  generator  circuit  by  a  condenser 
of  suitable  capacity,  thereby  generating  the  exciting  current  of 
the  motor  in  the  tertiary  circuit. 

The  primary  circuit  is  thereby  relieved  of  the  exciting  current 
of  the  motor,  the  efficiency  essentially  increased,  and  the  power- 
factor  of  the  single-phase  motor  with  condenser  in  tertiary  cir- 
cuit becomes  practically  unity  over  the  whole  range  of  load. 
At  the  same  time,  since  the  condenser  current  is  derived  by  double 


354     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

transformation  in  the  multitooth  structure  of  the  induction 
machine,  which  has  a  practically  uniform  magnetic  field,  irre- 
spective of  the  shape  of  the  primary  impressed  e.m.f.  wave,  the 
application  of  the  condenser  becomes  feasible  irrespective  of 
the  wave  shape  of  the  generator. 

Usually  the  tertiary  circuit  in  this  case  is  arranged  on  an  angle 
of  60  deg.  with  the  primary  circuit,  and  in  starting  a  powerful 
torque  is  thereby  developed,  with  a  torque  efficiency  superior  to 
any  other  single-phase  motor  starting  device,  and  when  com- 
bined with  inductive  reactance  in  a  second  tertiary  circuit,  the 
apparent  starting  torque  efficiency  can  be  made  even  to  exceed 
that  of  the  polyphase  induction  motor  (see  page  336). 

For  further  discussion  hereof,  see  A.  I.  E.  E.  Transactions, 
1900,  page  37. 

VH.  Frequency  Converter  or  General  Alternating-current 
Transformer 

159.  The  e.m.fs.  generated  in  the  secondary  of  the  induction 
machine  are  of  the  frequency  of  slip,  that  is,  synchronism  minus 
speed,  thus  of  lower  frequency  than  the  impressed  e.m.f.  in  the 
range  from  standstill  to  double  synchronism;  of  higher  frequency 
outside  of  this  range. ' 

Thus,  by  opening  the  secondary  circuits  of  the  induction 
machine  and  connecting  them  to  an  external  or  consumer's  cir- 
cuit, the  induction  machine  can  be  used  to  transform  from  one 
frequency  to  another,  as  frequency  converter. 

It  lowers  the  frequency  with  the  secondary  running  at  a  speed 
between  standstill  and  double  synchronism,  and  raises  the  fre- 
quency with  the  secondary  either  driven  backward  or  above 
double  synchronism. 

Obviously,  the  frequency  converter  can  at  the  same  time 
change  the  e.m.f.  by  using  a  suitable  number  of  primary  and 
secondary  turns,  and  can  change  the  phases  of  the  system  by 
having  a  secondary  wound  for  a  different  number  of  phases  from 
the  primary,  as,  for  instance,  convert  from  three  phase  6000 
volts  25  cycles  to  quarter  phase  2500  volts  62.5  cycles. 

Thus,  a  frequency  converter  can  be  called  a  "general  alter- 
nating-current transformer." 

For  its  theoretical  discussion  and  calculation,  see  "  Theory  and 
Calculation  of  Alternating-current  Phenomena." 

The  action  and  the  equations  of  the  general  alternating-current 


INDUCTION  MACHINES  355 

transformer  are  essentially  those  of  the  stationary  alternating- 
current  transformer,  except  that  the  ratio  of  secondary  to  primary 
generated  e.m.f .  is  not  the  ratio  of  turns  but  the  ratio  of.  the 
product  of  turns  and  frequency,  while  the  ratio  of  secondary 
current  and  primary  load  current  (that  is,  total  primary  current 
minus  primary  exciting  current)  is  the  inverse  ratio  of  turns. 

The  ratio  of  the  products  of  generated  e.m.f.  and  current,  that 
is,  the  ratio  of  electric  power  generated  in  the  secondary  to 
electric  power  consumed  in  the  primary  (less  excitation),  is  thus 
not  unity  but  is  the  ratio  of  secondary  to  primary  frequency. 

Hence,  when  lowering  the  frequency  with  the  secondary  re- 
volving at  a  speed  between  standstill  and  synchronism,  the 
secondary  output  is  less  than-  the  primary  input,  and  the  differ- 
ence is  transformed  into  mechanical  work;  that  is,  the  machine 
acts  at  the  same  time  as  induction  motor,  and  when  used  in  this 
manner  is  usually  connected  to  a  synchronous  or  induction  gen- 
erator feeding  preferably  into  the  secondary  circuit  (to  avoid 
double  transformation  of  its  output)  or  to  a  synchronous  con- 
verter, which  transforms  the  mechanical  power  of  the  frequency 
converter  into  electrical  power. 

When  raising  the  frequency  by  backward  rotation,  the  sec- 
ondary output  is  greater  than  the  primary  input  (or  rather  the 
electric  power  generated  in  the  secondary  greater  than  the  pri- 
mary power  consumed  by  the  generated  e.m.f.),  and  the  differ- 
ence is  to  be  supplied  by  mechanical  power  by  driving  the  fre- 
quency changer  backward  by  synchronous  or  induction  motor, 
preferably  connected  to  the  primary  circuit,  or  by  any  other 
motor  device. 

Above  synchronism  the  ratio  of  secondary  output  to  primary 
input  becomes  negative;  that  is,  the  induction  machine  generates 
power  in  the  primary  as  well  as  in  the  secondary,  the  primary 
power  at  the  impressed  frequency,  the  secondary  power  at  the 
frequency  of  slip,  and  thus  requires  mechanical  driving  power. 

The  secondary  power  and  frequency  are  less  than  the  primary 
below  double  synchronism,  more  above  double  synchronism, 
and  are  equal  at  double  synchronism,  so  that  at  double  syn- 
chronism the  primary  and  secondary  may  be  connected  in  multi- 
ple or  in  series  and  the  machine  is  then  a  double  synchronous 
alternator  further  discussed  in  the  "Theory  and  Calculation  of 
Electrical  Apparatus." 

As  far  as  its  transformer  action  is  concerned,  the  frequency 


356     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

converter  is  an  open  magnetic  circuit  transformer,  that  is,  a  trans- 
former of  relatively  high  magnetizing  current.  It  combines 
therewith,  however,  the  action  of  an  induction  motor  or  generator. 
Excluding  the  case  of  over-synchronous  rotation,  it  is  approxi- 
mately (that  is,  neglecting  internal  losses)  electrical  input  -r- 
electrical  output  -f-  mechanical  output  =  primary  frequency  -f- 
secondary  frequency  -f-  speed  or  primary  minus  secondary  fre- 
quency; that  is,  the  mechanical  output  is  negative  when  increas- 
ing the  frequency  by  backward  rotation. 

Such  frequency  converters  are  to  a  certain  extent  in  com- 
mercial use,  and  have  the  advantage  over  the  motor-generator 
plant  of  requiring  an  amount  of  apparatus  equal  only  to  the  out- 
put, while  the  motor-generator  set  requires  machinery  equal  to 
twice  the  output. 

An  application  of  the  frequency  converter  when  lowering  the 
frequency  is  made  in  concatenation  or  tandem  control  of  induc- 
tion machines,  as  described  in  the  next  section.  In  this  case 
the  first  motor,  or  all  the  motors  except  the  last  of  the  series  are 
in  reality  frequency  converters. 

VIII.  Concatenation  of  Induction  Motors 

160.  In  the  secondary  of  the  induction  motor  an  e.m.f.  is 
generated  of  the  frequency  of  slip.  Thus  connecting  the  sec- 
ondary circuit  of  the  induction  motor  to  the  primary  of  a  second 
induction  motor,  the  latter  is  fed  by  a  frequency  equal  to  the  slip 
of  the  first  motor,  and  reaches  its  synchronism  at  the  frequency 
of  slip  of  the  first  motor,  the  first  motor  then  acting  as  frequency 
converter  for  the  second  motor. 

If,  then,  two  equal  induction  motors  are  rigidly  connected 
together  and  thus  caused  to  revolve  at  the  same  speed,  the  speed 
of  the  second  motor,  which  is  the  slip  s  of  the  first  motor  at  no 
load,  equals  the  speed  of  the  first  motor:  s  =  1  —  s,  and  thus 
s  =  0.5.  That  is,  a  pair  of  induction  motors  connected  this  way 
in  tandem  or  in  concatenation,  that  is,  "  chain  connection/'  as 
commonly  called,  or  in  cascade,  as  called  abroad,  tends  to  ap- 
proach s  =  0.5,  or  half  synchronism,  at  no  load,  slipping  below 
this  speed  under  load;  that  is,  concatenation  of  two  motors  re- 
duces their  synchronous  speed  to  one-half,  and  thus  offers  as  means 
to  operate  induction  motors  at  one-half  speed. 

In  general,  if  a  number  of  induction  machines  are  connected 


INDUCTION  MACHINES  357 

in  tandem,  that  is,  the  secondary  of  each  motor  feeding  the 
primary  of  the  next  motor,  the  secondary  of  this  last  motor  being 
short-circuited,  the  sum  of  the  speeds  of  all  motors  tends  toward 
synchronism,  and  with  all  motors  connected  together  so  as  to 

revolve  at  the  same  speed  the  system  operates  at  -  synchronous 

speed,  when  n  =  number  of  motors.  If  the  two  induction 
motors  on  the  same  shaft  have  a  different  number  of  poles,  they 
synchronize  at  some  other  speed  below  synchronism,  or  if  con- 
nected differentially,  they  synchronize  at  some  speed  above 
synchronism. 

Assuming  the  ratio  of  turns  of  primary  and  secondary  as  1 : 1, 
with  two  equal  induction  motors  in  concatenation  at  standstill, 
the  frequency  and  the  e.m.f.  'impressed  upon  the  second  motor, 
neglecting  the  drop  of  e.m.f.  in  the  internal  impedance  of  the  first 
motor,  equal  those  of  the  first  motor.  With  increasing  speed, 
the  frequency  and  the  e.m.f.  impressed  upon  the  second  motor 
decrease  proportionally  to  each  other,  and  thus  the  magnetic 
flux  and  the  magnetic  density  in  the  second  motor,  and  its  ex- 
citing current,  remain  constant  and  equal  to  those  of  the  first 
motor,  neglecting  internal  losses;  that  is,  when  connected  in  con- 
catenation the  magnetic  density,  current  input,  etc.,  and  thus 
the  torque  developed  by  the  second  motor,  are  approximately 
equal  to  those  of  the  first  motor,  being  less  because  of  the  internal 
losses  in  the  first  motor. 

Hence,  the  motors  in  concatenation  share  the  work  in  approxi- 
mately equal  portions,  and  the  second  motor  utilizes  the  power 
which  without  the  use  of  a  second  motor  at  less  than  one-half 
synchronous  speed  would  have  to  be  wasted  in  the  secondary 
resistance;  that  is,  theoretically  concatenation  doubles  the  torque 
and  output  for  a  given  current,  or  power  input  into  the  motor 
system.  In  reality  the  gain  is  somewhat  less,  due  to  the  second 
motor  not  being  quite  equal  to  a  non-inductive  resistance  for 
the  secondary  of  the  first  motor,  and  due  to  the  drop  of  voltage 
in  the  internal  impedance  of  the  first  motor,  etc. 

At  one-half  synchronism,  that  is,  the  limiting  speed  of  the  con- 
catenated couple,  the  current  input  in  the  first  motor  equals  its 
exciting  current  plus  the  transformed  exciting  current  of  the 
second  motor,  that  is,  equals  twice  the  exciting  current. 

161.  Henee,  comparing  the  concatenated  couple  with  a  single 
motor,  the  primary  exciting  admittance  is  doubled.  The  total 


358     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

impedance,  primary  plus  secondary,  is  that  of  both  motors, 
that  is,  doubled  also,  and  the  characteristic  constant  of  the  con- 
catenated couple  is  thus  four  times  that  of  a  single  motor,  but 
the  speed  reduced  to  one-half. 


FIG.  192. — Comparison  of  concatenated  motors  with  a  single  motor  of 
double  the  number  of  poles. 

Comparing  the  concatenated  couple  with  a  single  motor  re- 
wound for  twice  the  number  of  poles,  that  is,  one-half  speed 
also,  such  rewinding  does  not  change  the  self-inductive  impe- 


INDUCTION  MACHINES 


359 


dance,  but  quadruples  the  exciting  admittance,  since  one-half 
as  many  turns  per  pole  have  to  produce  the  same  flux  in  one-half 
the  pole  arc,  that  is,  with  twice  the  density.  Thus  the  character- 
istic constant  is  increased  fourfold  also.  It  follows  herefrom 
that  the  characteristic  constant  of  the  concatenated  couple  is 
that  of  one  motor  rewound  for  twice  the  number  of  poles. 

The  slip  under  load,  however,  is  less  in  the  concatenated 
couple  than  in  the  motor  with  twice  the  number  of  poles,  being 
due  to  only  one-quarter  the  internal  impedance,  the  secondary 
impedance  of  the  second  motor  only,  and  thus  the  efficiency  is 
slightly  higher. 


1.3    1.2    1.1    1.0    0.9    0.8    0.7    0.6    0.5    0.4   0.3   0.2    0.1      0  -0.1-0.2-0.3-0.4-0.5-0.0-0.7 

FIG.  193. — Concatenation  of  induction  motors,  speed  curves. 

Two  motors  coupled  in  concatenation  are  in  the  range  from 
standstill  to  one-half  synchronism  approximately  equivalent 
to  one  motor  of  twice  the  admittance,  three  times  the  primary 
impedance,  and  the  same  secondary  impedance  as  each  of  the 
two  motors,  or  more  nearly  2.8  times  the  primary  and  1.2  times 
the  secondary  impedance  of  one  motor.  Such  motor  is  called 
the  equivalent  motor. 

162.  The  calculation  of  the  characteristic  curve  of  the  concate- 
nated motor  system  is  similar  to,  but  more  complex  than,  that 
of  the  single  motor.  Starting  from  the  generated  e.m.f.  e  of  the 
second  motor,  reduced  to  full  frequency,  we  work  up  to  the  im- 


360     ELEMENTS  OF  ELECTRICAL  ENGINEERING 


pressed  e.m.f.  of  the  first  motor  e0,  by  taking  due  consideration 
of  the  proper  frequencies  of  the  different  circuits.  Herefor  the 
reader  must  be  referred  to  "  Theory  and  Calculation  of  Electrical 
Apparatus." 

The  load  curves  of  the  pair  of  three-phase  motors  of  the  same 
constants  as  the  motor  in  Figs.  176  and  177  are  given  in  Fig.  192, 
the  complete  speed  curve  in  Fig.  193. 

Fig.  192  shows  the  load  curve  of  the  total  couple,  of  the  two 
individual  motors,  and  of  the  equivalent  motor. 

As  seen  from  the  speed  curve,  the  torque  from  standstill  to 
one-half  synchronism  has  the  same  shape  as  the  torque  curve  of 
a  single  motor  between  standstill  and  synchronism.  At  one-half 
synchronism  the  torque  reverses  and  becomes  negative.  It 
reverses  again  at  about  two-thirds  synchronism,  and  is  positive 


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FIG.  194. — Concatenation  of  induction  motors  speed  curve  with  resistance 
in  the  secondary  circuit. 

between  about  two-thirds  synchronism  and  synchronism,  zero 
at  synchronism,  and  negative  beyond  synchronism. 

Thus,  with  a  concatenated  couple,  two  ranges  of  positive 
torque  and  power  as  induction  motor  exist,  one  from  standstill 
to  half  synchronism,  the  other  from  about  two-thirds  synchro- 
nism to  synchronism. 

In  the  ranges  from  one-half  synchronism  to  about  two-thirds 
synchronism,  and  beyond  synchronism,  the  torque  is  negative, 
that  is,  the  couple  acts  as  generator. 

The  insertion  of  resistance  in  the  secondary  of  the  second 
motor  has  in  the  range  from  standstill  to  half  synchronism  the 
same  effect  asin  a  single  induction  motor,  that  is,  shifts  the  maxi- 
mum torque  point  toward  lower  speed  without  changing  its 
value.  Beyond  half  synchronism,  however,  resistance  in  the 


INDUCTION  MACHINES  361 

secondary  lengthens  the  generator  part  of  the  curve,  and  makes 
the  second  motor  part  of  the  curve  more  or  less  disappear,  as 
seen  in  Fig.  194,  which  gives  the  speed  curves  of  the  same  motor 
as  Fig.  193,  with  resistance  in  circuit  in  the  secondary  of  the 
second  motor. 

The  main  advantages  of  concatenation  are  obviously  the  abil- 
ity of  operating  at  two  different  speeds,  the  increased  torque  and 
power  efficiency  below  half  speed,  and  the  generator  or  braking 
action  between  half  speed  and  synchronism,  and  such  concatena- 
tion is  therefore  used  to  some  extent  in  three-phase  railway 
motor  equipments,  while  for  stationary  motors  usually  a  change 
of  the  number  of  poles  by  reconnecting  the  primary  winding 
through  a  suitable  switch  is  preferred  where  several  speeds  are 
desired,  as  it  requires  only  one  motor. 


INDEX 


Acceleration  with  starting  device  of 
single-phase  induction 

motor,  338 
Active  component,  40 

electromotive  force,  39 
Acyclic  generator,  11 

machine,  124 
Adjustable  speed  alternating-current 

commutator  motor,  221 
Admittance,  98 

exciting,  of  induction  motor,  311 
Air-blast  transformer,  295 
Air  gap  hi  reactor,  303 
Air  reactors,  305 
All    day    efficiency  of    transformer, 

284 
Alternating-current         commutator 

motor,  218 
generator    electromotive    force, 

16 

Angle  of  lag,  34 
Apparent  power  efficiency,  314 

torque  efficiency,  314 
Armature    current    of    synchronous 

converter,  233 
reaction  coefficient,  208 

of  commutating  machine,  193 

of  synchronous  converter,  245 

of  synchronous  machine,  130 

resistance  in  induction  motor, 

322,  333 
windings,  168 
Asynchronous  generator  and  motor, 

see  "induction" 
Attenuation  constant,  25 
Auto-transformer,  124,  299 

with    direct-current    converter, 

263 

with  three-wire  machines,  271 
Average  e.m.f.,  13 

Boosters,  122 

induction,  349 


Brush    shift    of    commutating    ma- 
chine, 181 

Capacity,  54 

Characteristic  constant  of  induction 

motor,  321,  330 
magnetic       of       commutating 

machine,  194 

Charging  current  of  condenser,  56 
Choke  coil,  124 

Circuit,  electric  and  magnetic,  2 
Closed  circuit  winding,  171 
Coefficient  of  armature  reaction,  208 
Commutating  field,  alternating-cur- 
rent motor,  220 
machines,  121,  166 
poles,  184 
Commutation,  199 

alternating-current  motor,  219 
Compensated       alternating-current 

series  motor,  221 

Compensating  winding  of  commu- 
tating machine,  190 
Compensation    for  power-factor  by 
alternating-current      com- 
mutator motor,  221 
Compensator,  124 

also  see  "auto-transformer" 
synchronous,  123 
Compound  commutating  machines, 

166 

generator,  213 
motor,  216 

Compounding  action  of  commutat- 
ing poles,  188 
curve  of  commutating  machine, 

196 
of  synchronous  machine,  139, 

144 

of  synchronous  converter,  250 
of  transmission  line,  90 
Concatenation  of  induction  motors, 
256 


363 


364 


INDEX 


Condensers,  54.  124 

starting   device  of  single-phase 

induction  motor,  338 
Condensive  reactance,  55 
Conductance,  100 

Constant,    characteristic    of    induc- 
tion motor,  321,  330 
Converter,  122 

direct-current,  262 

synchronous,  223 

three-Wire,  270 
Core  loss  of  transformer,  279 

type  transformer,  295 
Counter  e.m.f.  of  impedance,  35 

of  inductance,  32 

of  resistance,  33 
Crank  diagram,  41 
Cross  currents  in  synchronous  ma- 
chines, 155 

Cumulative  compounding,  166,  217 
Current,  electric,  9 

magnetic  field,  2 

ratio  of  converter,  230 

Delta  connection,  127 

of  transformer,  297 
current,  127 
voltage,  127 

Demagnetization  curve  of  separately 
excited    or    magneto    ma- 
chine, 209 
Demagnetizing    armature    reaction, 

181 

Diagram,  crank  or  vector,  40 
Dielectric  field,  113 
force,  113 
hysteresis,  56 
quantities,  116 

Differential  compounding,  166,  218 
Direct-current  converter,  262 

generator  e.m.f.,  14 
Distorting  armature  reaction,  181 
Distortion   of  flux  and  saturation, 

183 

by  armature  reaction,  181 
Division  of  load  in  parallel  opera- 
tion   of    synchronous  ma- 
chines, 154 
Double-current  generator,  123,  259 


Double  reentrant  winding,  171 

spiral  winding,  171 
Drum  winding,  168 
Dynamotors,  122 

Eddy  currents,  52 

losses  due  to  slots,  192 
Effective  reactance,  51 

resistance,  48 

values,  15 

Effect  of  inductance,  28 
Efficiency,  314 

commutating  machines,  198 

reactors,  302 

synchronous  machines,  149 

transformer,  280 

all-day,  284 
Electrical  quantities,  115 

symbols,  119 
Electric  circuit,  2 

current,  magnetic  field,  2 
Electrifying  force,  117 
Electrolytic  apparatus,  122 
Electromotive  force,  113 

consumed  by  impedance,  35 
by  inductance,  33 
by  resistance,  33 

generated,  12 
Electrostatic  apparatus,  122 

see  "dielectric" 

Elimination  of  higher  harmonics,  127 
Energy,  magnetic,  28 
Equipotential  surfaces,  115 
Equivalent  motor  of  concatenated 
induction  motor,  358 

sine  waves,  106 
Excitation  of  transformer,  279 
Exciting    admittance    of    induction 
motor,  311 

current,  49 

Field    characteristic,    commutating 

machine,  198 
commutating,  185 
of  dielectric  force,  113 
of  force,  112 

of  gravitational  force,  110 
intensity,  116 
magnetic,  1,  5 


INDEX 


865 


Field,  magnetic,  1 

of  magnetic  force,  113. 
Fluctuating    cross    current  in  syn- 
chronous machines,  155 
Flux,  magnetic,  1 

of  commutating  machine,  178 
Force,  fields  of,  112 

lines,  115 

mechanical  in  transformer,  294 
Form  factor,  126 

of  wave,  16 
Foucault  currents,  52 
Fractional  pitch  winding,  175 
Frequency  of  commutation,  200 

converters,  124,  354 

of  synchronous  converters,  257 
Friction,  molecular  magnetic,  49 
Full-pitch  winding,  175 

General    alternating-current    trans- 
former, 354 

symbols,  120 

wave,  107 
Generated  e.m.f.,  12 

of  synchronous  machine,  12& 
Gradient,  117 

magnetic,  3 
Gravimotive  force,  113 

Harmonics,  higher,  127 

third  in  transformer,  298 
Heating  in  direct-current  converter, 
269 

of  synchronous  converter,  233 

of  transformer,  294 
Higher  harmonics,  127 
High-frequency  cross  current  in  syn- 
chronous machines,  159 
High  reactance  transformer,  292 
Hysteresis,  48 

current,  50 

dielectric,  56 

lead,  50 

resistance,  54 

Impedance,  34,  98 

counter  e.m.f.  of,  35 
curves  of  transformer,  286 
e.m.f.  consumed  by,  35 


Impedance,  of    induction   booster, 
350 

of  induction  motor,  311 

of  transmission  line,  57 
Inductance,  21 

counter  e.m.f.,  32 

e.m.f.  consumed  by,  33 
Inductive  devices,    starting    single- 
phase  induction  motor,  334 

load,  transmission  line,  87 

reactance,  32 
Induction  apparatus,  stationary,  122 

booster,  349 

generator,  340,  343 

machines,  122,  306 

magnetic,  5 

Instantaneous  e.m.f.,  13 
Intensity  of  field,  113,  116 

magnetic,  1,  5 
Interpole,  186 
Inverted  converters,  123,  255 

Lag  angle,  34 
Lap  winding,  172 
Lead,  hysteretic,  50 
Leading  current  as  induction  gen- 
erator load,  347 

Leakage  flux  of  transformer,  285 
Level  surfaces,  115 
Line,  also  see  "transmission  line" 
of  force,  115 
reactance,  36 
Load  characteristic,  series  generator, 

213 

shunt  generator,  210 
transmission  line,  85 
curves  of  induction  generator, 

341 

of  induction  motor,  317,  329 
of  synchronous  machine,  141, 

145 
losses  in  synchronous  machines. 

150 

saturation  curve  of  commutat- 
ing machines,  196 
Loaded  transmission  line,  57 
Loss,  commutating  machines,  198 
synchronous  machines,  149 
transformer,  280 


366 


INDEX 


Low  reactance  transformer,  292 

Magnetic  characteristic  of  synchron- 
ous machine,  147 

circuit,  2 

energy,  28 

field,  1,  113 

flux,  1 

friction,  molecular,  49 

gradient,  3 

hysteresis,  49 

pole,  1 

reaction,  10 

symbols,  119 
Magnetization  curves,  8 

of  commutating  machine,  194 
Magnetizing  current,  49 

force,  3 
Magneto  generator,  209 

machines,  166 

Magnetomotive  force,  2,  113 
Maximum  e.m.f.,  13 

power,  transmission  line,  86 
Mechanical  force,  113 

in  transformer,  294 
Mohocyclic  starting  devices,  single- 
phase  induction  motor,  334 
Motive  force,  113 
Motor  converter,  261 

synchronous,  141 
Multiple,  also  see  "parallel" 

drum  winding,  168 

reentrant  winding,  172 

ring  winding,  168 

spiral  winding,  171 
Mutual  inductance,  21 

Neutral  of  commutating  machine, 

180 

Nomenclature,  118 
Nominal  generated  e.m.f.,  128 
Non-polar  machine,  11,  124 

Oil  circulation  in  transformer,  295 
cooled  transformer,  294 

Open-coil  winding,  171 

Output  of  direct-current  converter, 

269 
of  synchronous  converter,  238 


Over-compensation  of  armature  re- 
action, 190 

Over-compounding    curve,    commu- 
tating machine,  197 
of  transmission  line,  93 

Parallel  connection  of  circuits,  101 
operation  of  induction  genera- 
tor, 348 

of  synchronous  machines,  153 
Parallelogram  of  sine  wave,  44 
Percentage  saturation,  147,  195 
Permeability,  5 
Permittivity,  117 
Phase,  45 

characteristics   of   synchronous 

machines,  145 

control    by    synchronous    con- 
verter, 252 

of  transmission  line,  90 
converter,  123 

induction,  350 

splitting  devices,  starting  single- 
phase  induction  motor,  334 
Pitch  deficiency  and  wave  shape,  128 

of  winding,  175 
Polarization  cell,  124 ' 
Pole,  magnetic,  1 

strength,  magnetic,  1 
Polygon  of  sine  waves,  44 
Polyphase  alternating-current  com- 
mutator motor,  220 
induction  motor,  310 
Potential  regulator,  124 
Power,  15,  39 

component,  40 

efficiency,  314 

e.m.f.,  39 

factor     of     alternating-current 

commutator  motor,  219 
compensation  by  alternating- 
current  commutator  motor, 
221 

of  induction  generator,  343 
Pulsation  of  armature  reaction,  133 
of  flux  by  slots,  191 

Quantity,  electric,  118 

Racing  of  inverted  converter,  257 


INDEX 


367 


Railway  induction-phase  converter, 

353 
Rating  of  direct-current   converter, 

269 

of  synchronous  converter,  238 
Ratio  of  transformer,  67,  277 
Reactance,  99,  302 
condensive,  55 
effective,  51 
inductive,  32 
synchronous,  129,  136 
of  transformer,  285 
of  transmission  line,  36 
Reaction,  armature,  of  commutating 

machine,  193 

of  synchronous  converter,  245 
of  synchronous  machine,  130 
magnetic,  10 
Reactive  coil,  124,  302 
component,  40 

currents    in    synchronous    con- 
verter, 250 
e.m.f.,  39 
Reactor,  124,  302 
Real  generated  e.m.f.,  128 
Rectangular  coordinates,  77 
Rectifying  apparatus,  122 
Reentrant  winding,  171 
Regulation  of  auto-transformer,  302 
curve,    commutating    machine, 

198 

synchronous  generator,  140 
transformer,  285 
transmission  line,  57 
Regulator,  voltage,  124 
Reluctance,  21 
Repulsion  motor,  221 

in  transformer,  294 
Resistance,  9,  99 

characteristic,  series  generator, 

213 

shunt  generator,  211 
commutation,  200 
counter  e.m.f.,  33 
effective,  48 
e.m.f.  consumed  by,  33 
leads,  alternating-current  motor, 

220 
in  commutation,  206 


Resistivity,  10 
Ring  connection,  127 

winding,  168 

Rotating  magnetic   field  of  induc- 
tion motor,  309 

Saturation    curve    of   commutating 

machine,  194 

of  synchronous  machine,  147 
effect  of  flux  in  commutating 

machine,  182 

in  synchronous  machine,  148 
factor  of  commutating  machine. 

195 

of  synchronous  machine,  147 
percentage,  147,  195 
Self-inductance,  21 

of  commutation,  201 
of  synchronous  machine,  133 
of  transformer,  285 
Self-inductive  impedance  of  induc- 
tion motor,  311 
Self-induction,  10 
Separately     excited     commutating 

machine,  166 
generator,  209 

Series  alternating-current  motor,  221 
commutating  machines,  166 
connection  of  circuit,  101 
drum  "winding,  168 
generator,  211 
lap  winding,  174 
motor,  216 

Shell-type  transformer,  295 
Shh\  of   brushes   of    commutating 

machine,  181 

Short  circuit  of  auto-transformer,  302 
currents  of  alternator,  160 

of  transformer,  293 
loss  of  transformer,  286 
Shunt-corn  mutating  machine,  166 
generator,  210 
motor,  215 

Silicon  steel,  hysteresis,  52 
Sine  waves,  equivalent,  106 
Single-phase    alternating -current 

commutator  motors,  220 
converter,  229 
induction  motor,  309,  326 


368 


INDEX 


Single-phase,  short  circuit  of  alter- 
nator,  162 

Six-phase  converter,  230 
Size  of  auto-transformer,  300 

of  reactor,  303 

Slip  of  frequency,  induction  genera- 
tor, 343 

of  induction  booster,  350 
of  induction  motor,  309 
Slots,  effect  on  magnetic  flux,  190 
Speed  characteristic  of  series  motor, 

216 

shunt  generator,  211 
shunt  motor,  215 
curves  of  induction  generator, 

341 

of  induction  motor,  317,  329 
of  inverted  converter,  256 
Spiral  winding,  double  and  multiple, 

171 

Split-pole  converter,  252 
Stability  of  synchronous  converter, 

258 

Star  connection,  127 
Starting  of  current,  24 

devices  of  single-phase  induction 

motor,  333 

of  induction  motor,  3^2 
of  synchronous  converter,  253 
of  synchronous  motor,  151 
Stopping  of  current,  26 
Susceptance,  100 
Symbols,  119 
Symbolic  method,  77 
Synchronous  commutating  machine, 

123 

condenser,  123 
converter,  223 
induction  generator  and  motor, 

349 

machines,  122,  126 
motor  starting,  151 
reactance,  129,  136 
watts,  313 

Terminal  voltage,  128 
Third  harmonic  in  transformer,  298 
Three-phase  transformer,  297 
Three-wire  converter,  270 
generator,  270 


Time  constant,  25 
Torque  curves  of  induction  motor, 
323,  332 

efficiency,  314 

of  induction  motor,  313 

maximum,  of  induction  motor, 

324 
Transformer,  66,  77,  122,  277 

general  alterating  current,  354 

neutral,   with    three-wire    con- 
verter, 275 
Transmission  line,  compounding,  90 

impedance,  57 

load  characteristic,  85 

over-compounding,  93 

phase  control,  90 

reactance,  36 
Two  circuit  single-phase  converter, 

229 
Types  of  transformers,  295 

Unbalancing  of  polyphase  synchron- 
ous machines,  150 
Unipolar  machine,  11,  124 
Unit  current,  2 

e.m.f.,  9 

magnet  pole,  1 

Variable  ratio  converter,  252 

Variation  of  converter  voltage  ratio, 
231 

Vector  diagram,  41 

Ventilation  of  transformer,  294 

Virtual  generated  e.m.f.,  128 

Voltage  commutation,  201 

ratio  of  synchronous  converter, 
226 

Volt    ampere    characteristic    of    re- 
actor, 304 

Water  circulation  in  transformer,  295 
Wattless  component,  40 

e.m.f.,  39 

Wave  winding,  172 
Weight  of  mass,  113 

Y  connection,  127 

of  transformers,  297 
current,  127 
voltage,  127 

Zero  vector,  45 


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